Theorem keephyp2v 3718

Description: Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3704). (Contributed by NM, 16-Apr-2005.)

Hypotheses

Ref	Expression
keephyp2v.1	⊢ (A = if(φ, A, C) → (ψ ↔ χ))
keephyp2v.2	⊢ (B = if(φ, B, D) → (χ ↔ θ))
keephyp2v.3	⊢ (C = if(φ, A, C) → (τ ↔ η))
keephyp2v.4	⊢ (D = if(φ, B, D) → (η ↔ θ))
keephyp2v.5	⊢ ψ
keephyp2v.6	⊢ τ

Assertion

Ref	Expression
keephyp2v	⊢ θ

Proof of Theorem keephyp2v

Step	Hyp	Ref	Expression
1		keephyp2v.5	. . 3 ⊢ ψ
2		iftrue 3669	. . . . . 6 ⊢ (φ → if(φ, A, C) = A)
3	2	eqcomd 2358	. . . . 5 ⊢ (φ → A = if(φ, A, C))
4		keephyp2v.1	. . . . 5 ⊢ (A = if(φ, A, C) → (ψ ↔ χ))
5	3, 4	syl 15	. . . 4 ⊢ (φ → (ψ ↔ χ))
6		iftrue 3669	. . . . . 6 ⊢ (φ → if(φ, B, D) = B)
7	6	eqcomd 2358	. . . . 5 ⊢ (φ → B = if(φ, B, D))
8		keephyp2v.2	. . . . 5 ⊢ (B = if(φ, B, D) → (χ ↔ θ))
9	7, 8	syl 15	. . . 4 ⊢ (φ → (χ ↔ θ))
10	5, 9	bitrd 244	. . 3 ⊢ (φ → (ψ ↔ θ))
11	1, 10	mpbii 202	. 2 ⊢ (φ → θ)
12		keephyp2v.6	. . 3 ⊢ τ
13		iffalse 3670	. . . . . 6 ⊢ (¬ φ → if(φ, A, C) = C)
14	13	eqcomd 2358	. . . . 5 ⊢ (¬ φ → C = if(φ, A, C))
15		keephyp2v.3	. . . . 5 ⊢ (C = if(φ, A, C) → (τ ↔ η))
16	14, 15	syl 15	. . . 4 ⊢ (¬ φ → (τ ↔ η))
17		iffalse 3670	. . . . . 6 ⊢ (¬ φ → if(φ, B, D) = D)
18	17	eqcomd 2358	. . . . 5 ⊢ (¬ φ → D = if(φ, B, D))
19		keephyp2v.4	. . . . 5 ⊢ (D = if(φ, B, D) → (η ↔ θ))
20	18, 19	syl 15	. . . 4 ⊢ (¬ φ → (η ↔ θ))
21	16, 20	bitrd 244	. . 3 ⊢ (¬ φ → (τ ↔ θ))
22	12, 21	mpbii 202	. 2 ⊢ (¬ φ → θ)
23	11, 22	pm2.61i 156	1 ⊢ θ

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)