Theorem elimif 4560

Description: Elimination of a conditional operator contained in a wff 𝜓. (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.)

Hypotheses

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

Assertion

Ref	Expression
elimif	⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃)))

Proof of Theorem elimif

Step	Hyp	Ref	Expression
1		iftrue 4529	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2		elimif.1	. . 3 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝜓 ↔ 𝜒))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4		iffalse 4532	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
5		elimif.2	. . 3 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝜓 ↔ 𝜃))
6	4, 5	syl 17	. 2 ⊢ (¬ 𝜑 → (𝜓 ↔ 𝜃))
7	3, 6	cases 1039	1 ⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533  ifcif 4523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-if 4524

This theorem is referenced by:  eqif  4564  elif  4566  ifel  4567  ftc1anclem5  37076  clsk1indlem2  43350