Theorem elimhyp2v 3711

Description: Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)

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 3668	. . . . . 6 ⊢ (φ → if(φ, A, C) = A)
2	1	eqcomd 2358	. . . . 5 ⊢ (φ → A = if(φ, A, C))
3		elimhyp2v.1	. . . . 5 ⊢ (A = if(φ, A, C) → (φ ↔ χ))
4	2, 3	syl 15	. . . 4 ⊢ (φ → (φ ↔ χ))
5		iftrue 3668	. . . . . 6 ⊢ (φ → if(φ, B, D) = B)
6	5	eqcomd 2358	. . . . 5 ⊢ (φ → B = if(φ, B, D))
7		elimhyp2v.2	. . . . 5 ⊢ (B = if(φ, B, D) → (χ ↔ θ))
8	6, 7	syl 15	. . . 4 ⊢ (φ → (χ ↔ θ))
9	4, 8	bitrd 244	. . 3 ⊢ (φ → (φ ↔ θ))
10	9	ibi 232	. 2 ⊢ (φ → θ)
11		elimhyp2v.5	. . 3 ⊢ τ
12		iffalse 3669	. . . . . 6 ⊢ (¬ φ → if(φ, A, C) = C)
13	12	eqcomd 2358	. . . . 5 ⊢ (¬ φ → C = if(φ, A, C))
14		elimhyp2v.3	. . . . 5 ⊢ (C = if(φ, A, C) → (τ ↔ η))
15	13, 14	syl 15	. . . 4 ⊢ (¬ φ → (τ ↔ η))
16		iffalse 3669	. . . . . 6 ⊢ (¬ φ → if(φ, B, D) = D)
17	16	eqcomd 2358	. . . . 5 ⊢ (¬ φ → D = if(φ, B, D))
18		elimhyp2v.4	. . . . 5 ⊢ (D = if(φ, B, D) → (η ↔ θ))
19	17, 18	syl 15	. . . 4 ⊢ (¬ φ → (η ↔ θ))
20	15, 19	bitrd 244	. . 3 ⊢ (¬ φ → (τ ↔ θ))
21	11, 20	mpbii 202	. 2 ⊢ (¬ φ → θ)
22	10, 21	pm2.61i 156	1 ⊢ θ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   = wceq 1642   ifcif 3662

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 3663

This theorem is referenced by: (None)