Theorem elimdhyp 2392

Description: Version of elimhyp 2387 where the hypothesis is deduced from the final antecedent. See ghomgrplem 10345 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)

Hypotheses

Ref	Expression
elimdhyp.1	⊢ (φ → ψ)
elimdhyp.2	⊢ (A = if(φ, A, B) → (ψ ↔ χ))
elimdhyp.3	⊢ (B = if(φ, A, B) → (θ ↔ χ))
elimdhyp.4	⊢ θ

Assertion

Ref	Expression
elimdhyp	⊢ χ

Proof of Theorem elimdhyp

Step	Hyp	Ref	Expression
1		elimdhyp.1	. . 3 ⊢ (φ → ψ)
2		iftrue 2363	. . . . 5 ⊢ (φ → if(φ, A, B) = A)
3	2	eqcomd 1478	. . . 4 ⊢ (φ → A = if(φ, A, B))
4		elimdhyp.2	. . . 4 ⊢ (A = if(φ, A, B) → (ψ ↔ χ))
5	3, 4	syl 10	. . 3 ⊢ (φ → (ψ ↔ χ))
6	1, 5	mpbid 195	. 2 ⊢ (φ → χ)
7		elimdhyp.4	. . 3 ⊢ θ
8		iffalse 2364	. . . . 5 ⊢ (¬ φ → if(φ, A, B) = B)
9	8	eqcomd 1478	. . . 4 ⊢ (¬ φ → B = if(φ, A, B))
10		elimdhyp.3	. . . 4 ⊢ (B = if(φ, A, B) → (θ ↔ χ))
11	9, 10	syl 10	. . 3 ⊢ (¬ φ → (θ ↔ χ))
12	7, 11	mpbii 193	. 2 ⊢ (¬ φ → χ)
13	6, 12	pm2.61i 126	1 ⊢ χ

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

This theorem is referenced by:  ghomgrplem 10345

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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