Theorem elimhyp2v 1791

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 1780	. . . . . 6 ⊢ (φ → if(φ, A, C) = A)
2	1	cleqcomd 1106	. . . . 5 ⊢ (φ → A = if(φ, A, C))
3		elimhyp2v.1	. . . . 5 ⊢ (A = if(φ, A, C) → (φ ↔ χ))
4	2, 3	syl 12	. . . 4 ⊢ (φ → (φ ↔ χ))
5		iftrue 1780	. . . . . 6 ⊢ (φ → if(φ, B, D) = B)
6	5	cleqcomd 1106	. . . . 5 ⊢ (φ → B = if(φ, B, D))
7		elimhyp2v.2	. . . . 5 ⊢ (B = if(φ, B, D) → (χ ↔ θ))
8	6, 7	syl 12	. . . 4 ⊢ (φ → (χ ↔ θ))
9	4, 8	bitrd 406	. . 3 ⊢ (φ → (φ ↔ θ))
10	9	ibi 449	. 2 ⊢ (φ → θ)
11		elimhyp2v.5	. . 3 ⊢ τ
12		iffalse 1781	. . . . . 6 ⊢ (¬ φ → if(φ, A, C) = C)
13	12	cleqcomd 1106	. . . . 5 ⊢ (¬ φ → C = if(φ, A, C))
14		elimhyp2v.3	. . . . 5 ⊢ (C = if(φ, A, C) → (τ ↔ η))
15	13, 14	syl 12	. . . 4 ⊢ (¬ φ → (τ ↔ η))
16		iffalse 1781	. . . . . 6 ⊢ (¬ φ → if(φ, B, D) = D)
17	16	cleqcomd 1106	. . . . 5 ⊢ (¬ φ → D = if(φ, B, D))
18		elimhyp2v.4	. . . . 5 ⊢ (D = if(φ, B, D) → (η ↔ θ))
19	17, 18	syl 12	. . . 4 ⊢ (¬ φ → (η ↔ θ))
20	15, 19	bitrd 406	. . 3 ⊢ (¬ φ → (τ ↔ θ))
21	11, 20	mpbii 168	. 2 ⊢ (¬ φ → θ)
22	10, 21	pm2.61i 110	1 ⊢ θ

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   = wceq 1091  ifcif 1776

This theorem is referenced by:  hlimcau 5142  omls 5251  osumlem8 5537

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-if 1777

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