Theorem elimhyp2v 2388

Description: Eliminate a hypothesis containing 2 class variables.

Hypotheses

Ref	Expression
elimhyp2v.1	⊢ (A = if(φ, A, C) → (φ ↔ χ))
elimhyp2v.2	⊢ (B = if(φ, B, D) → (χ ↔ θ))
elimhyp2v.3	⊢ (C = if(φ, A, C) → (τ ↔ η))
elimhyp2v.4	⊢ (D = if(φ, B, D) → (η ↔ θ))
elimhyp2v.5	⊢ τ

Assertion

Ref	Expression
elimhyp2v	⊢ θ

Proof of Theorem elimhyp2v

Step	Hyp	Ref	Expression
1		iftrue 2363	. . . . . 6 ⊢ (φ → if(φ, A, C) = A)
2	1	eqcomd 1478	. . . . 5 ⊢ (φ → A = if(φ, A, C))
3		elimhyp2v.1	. . . . 5 ⊢ (A = if(φ, A, C) → (φ ↔ χ))
4	2, 3	syl 10	. . . 4 ⊢ (φ → (φ ↔ χ))
5		iftrue 2363	. . . . . 6 ⊢ (φ → if(φ, B, D) = B)
6	5	eqcomd 1478	. . . . 5 ⊢ (φ → B = if(φ, B, D))
7		elimhyp2v.2	. . . . 5 ⊢ (B = if(φ, B, D) → (χ ↔ θ))
8	6, 7	syl 10	. . . 4 ⊢ (φ → (χ ↔ θ))
9	4, 8	bitrd 527	. . 3 ⊢ (φ → (φ ↔ θ))
10	9	ibi 591	. 2 ⊢ (φ → θ)
11		elimhyp2v.5	. . 3 ⊢ τ
12		iffalse 2364	. . . . . 6 ⊢ (¬ φ → if(φ, A, C) = C)
13	12	eqcomd 1478	. . . . 5 ⊢ (¬ φ → C = if(φ, A, C))
14		elimhyp2v.3	. . . . 5 ⊢ (C = if(φ, A, C) → (τ ↔ η))
15	13, 14	syl 10	. . . 4 ⊢ (¬ φ → (τ ↔ η))
16		iffalse 2364	. . . . . 6 ⊢ (¬ φ → if(φ, B, D) = D)
17	16	eqcomd 1478	. . . . 5 ⊢ (¬ φ → D = if(φ, B, D))
18		elimhyp2v.4	. . . . 5 ⊢ (D = if(φ, B, D) → (η ↔ θ))
19	17, 18	syl 10	. . . 4 ⊢ (¬ φ → (η ↔ θ))
20	15, 19	bitrd 527	. . 3 ⊢ (¬ φ → (τ ↔ θ))
21	11, 20	mpbii 193	. 2 ⊢ (¬ φ → θ)
22	10, 21	pm2.61i 126	1 ⊢ θ

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   = wceq 955   ifcif 2358

This theorem is referenced by:  bcpasc2t 6921  cvgcmp3cetlem1 7141  cvgcmp3cetlem2 7142  hlimcau 9062  omls 9201  osumlem8 9542

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-if 2359

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