Theorem elif 4512

Description: Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)

Assertion

Ref	Expression
elif	⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶)))

Proof of Theorem elif

Step	Hyp	Ref	Expression
1		eleq2 2904	. 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴 ∈ 𝐵))
2		eleq2 2904	. 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴 ∈ 𝐶))
3	1, 2	elimif 4506	1 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶)))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∈ wcel 2113  ifcif 4470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-if 4471

This theorem is referenced by:  clsk1indlem3  40399