Theorem ifeqor 4524

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 4478	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	con3i 154	. . 3 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → ¬ 𝜑)
3	2	iffalsed 4483	. 2 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	orri 862	1 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)

Syntax hints:  ¬ wn 3   ∨ wo 847   = wceq 1541  ifcif 4472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-if 4473

This theorem is referenced by:  ifpr  4643  rabrsn  4674  prmolefac  16958  muval2  27071  abssor  28184  finxpreclem2  37434  relexpxpmin  43820