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Theorem fvifeq 43702
Description: Equality of function values with conditional arguments, see also fvif 6675. (Contributed by Alexander van der Vekens, 21-May-2018.)
Assertion
Ref Expression
fvifeq (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = if(𝜑, (𝐹𝐵), (𝐹𝐶)))

Proof of Theorem fvifeq
StepHypRef Expression
1 fveq2 6659 . 2 (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = (𝐹‘if(𝜑, 𝐵, 𝐶)))
2 fvif 6675 . 2 (𝐹‘if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐹𝐵), (𝐹𝐶))
31, 2syl6eq 2875 1 (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = if(𝜑, (𝐹𝐵), (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  ifcif 4450  cfv 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-iota 6303  df-fv 6352
This theorem is referenced by: (None)
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