Theorem elimdhyp 4151

Description: Version of elimhyp 4146 where the hypothesis is deduced from the final antecedent. See divalg 15126 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 4092	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3	2	eqcomd 2628	. . . 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 222	. 2 ⊢ (𝜑 → 𝜒)
7		elimdhyp.4	. . 3 ⊢ 𝜃
8		iffalse 4095	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
9	8	eqcomd 2628	. . . 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 223	. 2 ⊢ (¬ 𝜑 → 𝜒)
13	6, 12	pm2.61i 176	1 ⊢ 𝜒

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1483  ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  divalg  15126