Theorem elimdhyp 4526

Description: Version of elimhyp 4521 where the hypothesis is deduced from the final antecedent. See divalg 16040 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 4462	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3	2	eqcomd 2744	. . . 4 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐵))
4		elimdhyp.2	. . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜒))
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
6	1, 5	mpbid 231	. 2 ⊢ (𝜑 → 𝜒)
7		elimdhyp.4	. . 3 ⊢ 𝜃
8		iffalse 4465	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
9	8	eqcomd 2744	. . . 4 ⊢ (¬ 𝜑 → 𝐵 = if(𝜑, 𝐴, 𝐵))
10		elimdhyp.3	. . . 4 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃 ↔ 𝜒))
11	9, 10	syl 17	. . 3 ⊢ (¬ 𝜑 → (𝜃 ↔ 𝜒))
12	7, 11	mpbii 232	. 2 ⊢ (¬ 𝜑 → 𝜒)
13	6, 12	pm2.61i 182	1 ⊢ 𝜒

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1539  ifcif 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-if 4457

This theorem is referenced by:  divalg  16040