Theorem fvif 6679

Description: Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
fvif	⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵))

Proof of Theorem fvif

Step	Hyp	Ref	Expression
1		fveq2 6663	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐴))
2		fveq2 6663	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐵))
3	1, 2	ifsb 4478	1 ⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵))

Syntax hints:   = wceq 1531  ifcif 4465  ‘cfv 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356

This theorem is referenced by:  ccatco  14189  sumeq2ii  15042  prodeq2ii  15259  ruclem1  15576  xpsrnbas  16836  mat2pmat1  21332  decpmatid  21370  pmatcollpwscmatlem1  21389  copco  23614  pcopt  23618  pcopt2  23619  limccnp  24481  prmorcht  25747  pclogsum  25783  mblfinlem2  34922  ftc1anclem8  34966  ftc1anc  34967  fvifeq  43469