Theorem ifeqor 3700

Description: The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
ifeqor	⊢ ( if(φ, A, B) = A ∨ if(φ, A, B) = B)

Proof of Theorem ifeqor

Step	Hyp	Ref	Expression
1		iftrue 3669	. . . 4 ⊢ (φ → if(φ, A, B) = A)
2	1	con3i 127	. . 3 ⊢ (¬ if(φ, A, B) = A → ¬ φ)
3		iffalse 3670	. . 3 ⊢ (¬ φ → if(φ, A, B) = B)
4	2, 3	syl 15	. 2 ⊢ (¬ if(φ, A, B) = A → if(φ, A, B) = B)
5	4	orri 365	1 ⊢ ( if(φ, A, B) = A ∨ if(φ, A, B) = B)

Syntax hints:  ¬ wn 3   ∨ wo 357   = wceq 1642   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:  ifpr  3775  enprmaplem5  6081