Theorem ifeqor 4537

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 4491	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	con3i 155	. . 3 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → ¬ 𝜑)
3	2	iffalsed 4496	. 2 ⊢ (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	orri 876	1 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1570  ifcif 4485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-if 4486

This theorem is used by:  ifpr  4657  rabrsn  4688  prmolefac  17142  muval2  27371  abssor  28512  finxpreclem2  38146  relexpxpmin  44559