Theorem ifeqor 4587

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 4542	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	con3i 154	. . 3 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → ¬ 𝜑)
3	2	iffalsed 4547	. 2 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	orri 860	1 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)

Syntax hints:  ¬ wn 3   ∨ wo 845   = wceq 1534  ifcif 4536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-if 4537

This theorem is referenced by:  ifpr  4703  rabrsn  4736  prmolefac  17069  muval2  27185  abssor  28264  finxpreclem2  37144  relexpxpmin  43455