Theorem elif 3697

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 2414	. 2 ⊢ ( if(φ, B, C) = B → (A ∈ if(φ, B, C) ↔ A ∈ B))
2		eleq2 2414	. 2 ⊢ ( if(φ, B, C) = C → (A ∈ if(φ, B, C) ↔ A ∈ C))
3	1, 2	elimif 3692	1 ⊢ (A ∈ if(φ, B, C) ↔ ((φ ∧ A ∈ B) ∨ (¬ φ ∧ A ∈ C)))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∈ wcel 1710   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3664

This theorem is referenced by:  enprmaplem4  6080