Theorem ifval 4535

Description: Another expression of the value of the if predicate, analogous to eqif 4534. See also the more specialized iftrue 4498 and iffalse 4501. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
ifval	⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶)))

Proof of Theorem ifval

Step	Hyp	Ref	Expression
1		eqif 4534	. 2 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶)))
2		cases2 1063	. 2 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶)) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶)))
3	1, 2	bitri 278	1 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ifcif 4492

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-if 4493

This theorem is used by:  dfiota4  6535  bj-projval  37672  dfaiota3  47869