Theorem elimhyp3v 3713

Description: Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)

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 3669	. . . . . 6 ⊢ (φ → if(φ, A, D) = A)
2	1	eqcomd 2358	. . . . 5 ⊢ (φ → A = if(φ, A, D))
3		elimhyp3v.1	. . . . 5 ⊢ (A = if(φ, A, D) → (φ ↔ χ))
4	2, 3	syl 15	. . . 4 ⊢ (φ → (φ ↔ χ))
5		iftrue 3669	. . . . . 6 ⊢ (φ → if(φ, B, R) = B)
6	5	eqcomd 2358	. . . . 5 ⊢ (φ → B = if(φ, B, R))
7		elimhyp3v.2	. . . . 5 ⊢ (B = if(φ, B, R) → (χ ↔ θ))
8	6, 7	syl 15	. . . 4 ⊢ (φ → (χ ↔ θ))
9		iftrue 3669	. . . . . 6 ⊢ (φ → if(φ, C, S) = C)
10	9	eqcomd 2358	. . . . 5 ⊢ (φ → C = if(φ, C, S))
11		elimhyp3v.3	. . . . 5 ⊢ (C = if(φ, C, S) → (θ ↔ τ))
12	10, 11	syl 15	. . . 4 ⊢ (φ → (θ ↔ τ))
13	4, 8, 12	3bitrd 270	. . 3 ⊢ (φ → (φ ↔ τ))
14	13	ibi 232	. 2 ⊢ (φ → τ)
15		elimhyp3v.7	. . 3 ⊢ η
16		iffalse 3670	. . . . . 6 ⊢ (¬ φ → if(φ, A, D) = D)
17	16	eqcomd 2358	. . . . 5 ⊢ (¬ φ → D = if(φ, A, D))
18		elimhyp3v.4	. . . . 5 ⊢ (D = if(φ, A, D) → (η ↔ ζ))
19	17, 18	syl 15	. . . 4 ⊢ (¬ φ → (η ↔ ζ))
20		iffalse 3670	. . . . . 6 ⊢ (¬ φ → if(φ, B, R) = R)
21	20	eqcomd 2358	. . . . 5 ⊢ (¬ φ → R = if(φ, B, R))
22		elimhyp3v.5	. . . . 5 ⊢ (R = if(φ, B, R) → (ζ ↔ σ))
23	21, 22	syl 15	. . . 4 ⊢ (¬ φ → (ζ ↔ σ))
24		iffalse 3670	. . . . . 6 ⊢ (¬ φ → if(φ, C, S) = S)
25	24	eqcomd 2358	. . . . 5 ⊢ (¬ φ → S = if(φ, C, S))
26		elimhyp3v.6	. . . . 5 ⊢ (S = if(φ, C, S) → (σ ↔ τ))
27	25, 26	syl 15	. . . 4 ⊢ (¬ φ → (σ ↔ τ))
28	19, 23, 27	3bitrd 270	. . 3 ⊢ (¬ φ → (η ↔ τ))
29	15, 28	mpbii 202	. 2 ⊢ (¬ φ → τ)
30	14, 29	pm2.61i 156	1 ⊢ τ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   = wceq 1642   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3664

This theorem is referenced by: (None)