Theorem keephyp2v 2394

Description: Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 2380).

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 2363	. . . . . 6 ⊢ (φ → if(φ, A, C) = A)
3	2	eqcomd 1478	. . . . 5 ⊢ (φ → A = if(φ, A, C))
4		keephyp2v.1	. . . . 5 ⊢ (A = if(φ, A, C) → (ψ ↔ χ))
5	3, 4	syl 10	. . . 4 ⊢ (φ → (ψ ↔ χ))
6		iftrue 2363	. . . . . 6 ⊢ (φ → if(φ, B, D) = B)
7	6	eqcomd 1478	. . . . 5 ⊢ (φ → B = if(φ, B, D))
8		keephyp2v.2	. . . . 5 ⊢ (B = if(φ, B, D) → (χ ↔ θ))
9	7, 8	syl 10	. . . 4 ⊢ (φ → (χ ↔ θ))
10	5, 9	bitrd 527	. . 3 ⊢ (φ → (ψ ↔ θ))
11	1, 10	mpbii 193	. 2 ⊢ (φ → θ)
12		keephyp2v.6	. . 3 ⊢ τ
13		iffalse 2364	. . . . . 6 ⊢ (¬ φ → if(φ, A, C) = C)
14	13	eqcomd 1478	. . . . 5 ⊢ (¬ φ → C = if(φ, A, C))
15		keephyp2v.3	. . . . 5 ⊢ (C = if(φ, A, C) → (τ ↔ η))
16	14, 15	syl 10	. . . 4 ⊢ (¬ φ → (τ ↔ η))
17		iffalse 2364	. . . . . 6 ⊢ (¬ φ → if(φ, B, D) = D)
18	17	eqcomd 1478	. . . . 5 ⊢ (¬ φ → D = if(φ, B, D))
19		keephyp2v.4	. . . . 5 ⊢ (D = if(φ, B, D) → (η ↔ θ))
20	18, 19	syl 10	. . . 4 ⊢ (¬ φ → (η ↔ θ))
21	16, 20	bitrd 527	. . 3 ⊢ (¬ φ → (τ ↔ θ))
22	12, 21	mpbii 193	. 2 ⊢ (¬ φ → θ)
23	11, 22	pm2.61i 126	1 ⊢ θ

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   = wceq 955   ifcif 2358

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-if 2359

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