Theorem elimdhyp 3715

Description: Version of elimhyp 3710 where the hypothesis is deduced from the final antecedent. See ghomgrplem in set.mm 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 3668	. . . . 5 ⊢ (φ → if(φ, A, B) = A)
3	2	eqcomd 2358	. . . 4 ⊢ (φ → A = if(φ, A, B))
4		elimdhyp.2	. . . 4 ⊢ (A = if(φ, A, B) → (ψ ↔ χ))
5	3, 4	syl 15	. . 3 ⊢ (φ → (ψ ↔ χ))
6	1, 5	mpbid 201	. 2 ⊢ (φ → χ)
7		elimdhyp.4	. . 3 ⊢ θ
8		iffalse 3669	. . . . 5 ⊢ (¬ φ → if(φ, A, B) = B)
9	8	eqcomd 2358	. . . 4 ⊢ (¬ φ → B = if(φ, A, B))
10		elimdhyp.3	. . . 4 ⊢ (B = if(φ, A, B) → (θ ↔ χ))
11	9, 10	syl 15	. . 3 ⊢ (¬ φ → (θ ↔ χ))
12	7, 11	mpbii 202	. 2 ⊢ (¬ φ → χ)
13	6, 12	pm2.61i 156	1 ⊢ χ

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)