Theorem elimdhyp 4101

Description: Version of elimhyp 4096 where the hypothesis is deduced from the final antecedent. See divalg 14964 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 4042	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3	2	eqcomd 2616	. . . 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 221	. 2 ⊢ (𝜑 → 𝜒)
7		elimdhyp.4	. . 3 ⊢ 𝜃
8		iffalse 4045	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
9	8	eqcomd 2616	. . . 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 222	. 2 ⊢ (¬ 𝜑 → 𝜒)
13	6, 12	pm2.61i 175	1 ⊢ 𝜒

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   = wceq 1475  ifcif 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-if 4037

This theorem is referenced by:  divalg  14964