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Theorem fvifeq 42303
 Description: Equality of function values with conditional arguments, see also fvif 6462. (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 6446 . 2 (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = (𝐹‘if(𝜑, 𝐵, 𝐶)))
2 fvif 6462 . 2 (𝐹‘if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐹𝐵), (𝐹𝐶))
31, 2syl6eq 2829 1 (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹𝐴) = if(𝜑, (𝐹𝐵), (𝐹𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601  ifcif 4306  ‘cfv 6135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143 This theorem is referenced by: (None)
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