Theorem elimhyp3v 2388

Description: Eliminate a hypothesis containing 3 class variables.

Hypotheses

Ref	Expression
elimhyp3v.1	⊢ (A = if(φ, A, D) → (φ ↔ χ))
elimhyp3v.2	⊢ (B = if(φ, B, R) → (χ ↔ θ))
elimhyp3v.3	⊢ (C = if(φ, C, S) → (θ ↔ τ))
elimhyp3v.4	⊢ (D = if(φ, A, D) → (η ↔ ζ))
elimhyp3v.5	⊢ (R = if(φ, B, R) → (ζ ↔ σ))
elimhyp3v.6	⊢ (S = if(φ, C, S) → (σ ↔ τ))
elimhyp3v.7	⊢ η

Assertion

Ref	Expression
elimhyp3v	⊢ τ

Proof of Theorem elimhyp3v

Step	Hyp	Ref	Expression
1		iftrue 2362	. . . . . 6 ⊢ (φ → if(φ, A, D) = A)
2	1	eqcomd 1477	. . . . 5 ⊢ (φ → A = if(φ, A, D))
3		elimhyp3v.1	. . . . 5 ⊢ (A = if(φ, A, D) → (φ ↔ χ))
4	2, 3	syl 10	. . . 4 ⊢ (φ → (φ ↔ χ))
5		iftrue 2362	. . . . . 6 ⊢ (φ → if(φ, B, R) = B)
6	5	eqcomd 1477	. . . . 5 ⊢ (φ → B = if(φ, B, R))
7		elimhyp3v.2	. . . . 5 ⊢ (B = if(φ, B, R) → (χ ↔ θ))
8	6, 7	syl 10	. . . 4 ⊢ (φ → (χ ↔ θ))
9		iftrue 2362	. . . . . 6 ⊢ (φ → if(φ, C, S) = C)
10	9	eqcomd 1477	. . . . 5 ⊢ (φ → C = if(φ, C, S))
11		elimhyp3v.3	. . . . 5 ⊢ (C = if(φ, C, S) → (θ ↔ τ))
12	10, 11	syl 10	. . . 4 ⊢ (φ → (θ ↔ τ))
13	4, 8, 12	3bitrd 543	. . 3 ⊢ (φ → (φ ↔ τ))
14	13	ibi 591	. 2 ⊢ (φ → τ)
15		elimhyp3v.7	. . 3 ⊢ η
16		iffalse 2363	. . . . . 6 ⊢ (¬ φ → if(φ, A, D) = D)
17	16	eqcomd 1477	. . . . 5 ⊢ (¬ φ → D = if(φ, A, D))
18		elimhyp3v.4	. . . . 5 ⊢ (D = if(φ, A, D) → (η ↔ ζ))
19	17, 18	syl 10	. . . 4 ⊢ (¬ φ → (η ↔ ζ))
20		iffalse 2363	. . . . . 6 ⊢ (¬ φ → if(φ, B, R) = R)
21	20	eqcomd 1477	. . . . 5 ⊢ (¬ φ → R = if(φ, B, R))
22		elimhyp3v.5	. . . . 5 ⊢ (R = if(φ, B, R) → (ζ ↔ σ))
23	21, 22	syl 10	. . . 4 ⊢ (¬ φ → (ζ ↔ σ))
24		iffalse 2363	. . . . . 6 ⊢ (¬ φ → if(φ, C, S) = S)
25	24	eqcomd 1477	. . . . 5 ⊢ (¬ φ → S = if(φ, C, S))
26		elimhyp3v.6	. . . . 5 ⊢ (S = if(φ, C, S) → (σ ↔ τ))
27	25, 26	syl 10	. . . 4 ⊢ (¬ φ → (σ ↔ τ))
28	19, 23, 27	3bitrd 543	. . 3 ⊢ (¬ φ → (η ↔ τ))
29	15, 28	mpbii 193	. 2 ⊢ (¬ φ → τ)
30	14, 29	pm2.61i 126	1 ⊢ τ

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   = wceq 954   ifcif 2357

This theorem is referenced by:  climuni 7044  hlimuni 9048  projlem7 9131

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-if 2358

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