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Theorem fvifeq 40240
Description: Equality of function values with conditional arguments, see also fvif 5998. (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 5987 . 2 (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = (𝐹‘if(𝜑, 𝐵, 𝐶)))
2 fvif 5998 . 2 (𝐹‘if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐹𝐵), (𝐹𝐶))
31, 2syl6eq 2564 1 (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = if(𝜑, (𝐹𝐵), (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  ifcif 3939  cfv 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-iota 5653  df-fv 5697
This theorem is referenced by: (None)
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