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Q.E.D.

Q E.D.

1. a,b,c are arbitrary specific constants and a=b and b=c [OSC1]
2. ∀w[P(w) iff ~(w = c) [Df]
3. P(a) iff ~(a=c) [2;UI]
4. P(b) iff ~(b=c) [2;UI]
5. a=b [1; simplification]
6. If a = b & P(a) then P(b) [substitution]
7. If a=b & ~(a=c) then ~(b=c) [3,4,6; rule of replacement]
8. If a=b then (if ~(a=c) then ~(b=c) ) [7; EXPORTATION]
9. If a=b then (if b=c then a=c) [8;trans]
10. If a=b and b=c then a=c [9;IMPORTATION]
11. ∀A∀B∀C[if A=B & B=C then A=C], where A,B,C are specific constants [10;UG]

1. a,b,c are arbitrary specific constants. [OSC1]
2. ∀w[P(w) iff ~(w = c) [Df]
3. P(a) iff ~(a=c) [2;UI]
4. P(b) iff ~(b=c) [2;UI]
5. If a = b & P(a) then P(b) [substitution]
6. If a=b & ~(a=c) then ~(b=c) [3,4,5; rule of replacement]
7. If a=b then (if ~(a=c) then ~(b=c) ) [6; EXPORTATION]
8. If a=b then (if b=c then a=c) [7;trans]
9. If a=b and b=c then a=c [8;IMPORTATION]
10. ∀A∀B∀C[if A=B & B=C then A=C], where A,B,C are specific constants [9;UG]

Line 4 is true, and has a truth value that is constant in time, so it is necessarily true, by my understanding of modal logic.

5. ☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] ∨ P(x_i) } [4; modal logic]
6. ☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] } ∨ P(x_i) [5; modal logic]
7. ◊¬{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] }
8. ~☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] } [7;Df]
9. P(x_i) [6,8;DS]

Line 4 is true, and has a truth value that is constant in time, so it is necessarily true, by my understanding of temporal modal logic.

5. ☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] ∨ P(x_i) } [4; TML]

If it is possible that all the P(x_i)'s have a truth value that varies in time, then they could all simultaneously be false, in which case line 5 would be false. Therefore at least one of the P(x_i)'s has a truth value that is constant in time. Let it be P(x_i). Therefore

6. If ☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] ∨ P(x_i) } then ☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] } ∨ P(x_i) [5; TML]

7. ☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] } ∨ P(x_i) [5,6; MP]

It is possible that all the P(x_i)'s except P(x_i) have a truth value that varies in time, so they could all be simultaneously false. Therefore

8. ◊¬{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] }
9. ~☐{[[P(x1) ∨ P(x2) ∨... P(x_i-1)] ∨ [P(x_i+1 ∨...]] } [8;Df]
10. P(x_i) [7,9;DS]