Theorem fvifeq 47390

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 2782	1 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶)))

Syntax hints:   → wi 4   = wceq 1541  ifcif 4472  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489

This theorem is referenced by: (None)