Theorem fvifeq 47274

Description: Equality of function values with conditional arguments, see also fvif 6838. (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 6822	. 2 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = (𝐹‘if(𝜑, 𝐵, 𝐶)))
2		fvif 6838	. 2 ⊢ (𝐹‘if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶))
3	1, 2	eqtrdi 2780	1 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶)))

Syntax hints:   → wi 4   = wceq 1540  ifcif 4476  ‘cfv 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-iota 6438  df-fv 6490

This theorem is referenced by: (None)