Theorem fvifeq 47250

Description: Equality of function values with conditional arguments, see also fvif 6902. (Contributed by Alexander van der Vekens, 21-May-2018.)

Assertion

Ref	Expression
fvifeq	⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶)))

Proof of Theorem fvifeq

Step	Hyp	Ref	Expression
1		fveq2 6886	. 2 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = (𝐹‘if(𝜑, 𝐵, 𝐶)))
2		fvif 6902	. 2 ⊢ (𝐹‘if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶))
3	1, 2	eqtrdi 2785	1 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶)))

Syntax hints:   → wi 4   = wceq 1539  ifcif 4505  ‘cfv 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-iota 6494  df-fv 6549

This theorem is referenced by: (None)