Theorem ifeqor 4510

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 4465	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	con3i 154	. . 3 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → ¬ 𝜑)
3	2	iffalsed 4470	. 2 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	orri 859	1 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)

Syntax hints:  ¬ wn 3   ∨ wo 844   = wceq 1539  ifcif 4459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-if 4460

This theorem is referenced by:  ifpr  4627  rabrsn  4660  prmolefac  16747  muval2  26283  finxpreclem2  35561  relexpxpmin  41325