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Theorem fvifeq 45482
Description: Equality of function values with conditional arguments, see also fvif 6856. (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 6840 . 2 (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = (𝐹‘if(𝜑, 𝐵, 𝐶)))
2 fvif 6856 . 2 (𝐹‘if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐹𝐵), (𝐹𝐶))
31, 2eqtrdi 2792 1 (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = if(𝜑, (𝐹𝐵), (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  ifcif 4485  cfv 6494
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-iota 6446  df-fv 6502
This theorem is referenced by: (None)
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