Theorem fvif 6661

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 6645	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐴))
2		fveq2 6645	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐵))
3	1, 2	ifsb 4438	1 ⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵))

Syntax hints:   = wceq 1538  ifcif 4425  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332

This theorem is referenced by:  ccatco  14188  sumeq2ii  15042  prodeq2ii  15259  ruclem1  15576  xpsrnbas  16836  mat2pmat1  21337  decpmatid  21375  pmatcollpwscmatlem1  21394  copco  23623  pcopt  23627  pcopt2  23628  limccnp  24494  prmorcht  25763  pclogsum  25799  mblfinlem2  35095  ftc1anclem8  35137  ftc1anc  35138  fvifeq  43836