Theorem elimif 4072

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 4042	. . 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 4045	. . 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 1004	1 ⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = 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:  eqif  4076  elif  4078  ifel  4079  ftc1anclem5  32659  clsk1indlem2  37360