Theorem ifval 4508

Description: Another expression of the value of the if predicate, analogous to eqif 4507. See also the more specialized iftrue 4473 and iffalse 4476. (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 4507	. 2 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶)))
2		cases2 1042	. 2 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶)) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶)))
3	1, 2	bitri 277	1 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533  ifcif 4467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-if 4468

This theorem is referenced by:  dfiota4  6342  bj-projval  34303