Theorem fvif 6898

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 6882	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐴))
2		fveq2 6882	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐵))
3	1, 2	ifsb 4506	1 ⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵))

Syntax hints:   = wceq 1567  ifcif 4492  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545

This theorem is referenced by:  ccatco  14871  sumeq2ii  15743  prodeq2ii  15964  ruclem1  16286  xpsrnbas  17624  rhmmpl  22508  rhmply1vr1  22512  mat2pmat1  22857  decpmatid  22895  pmatcollpwscmatlem1  22914  copco  25145  pcopt  25149  pcopt2  25150  limccnp  26018  prmorcht  27307  pclogsum  27344  esplyfval0  33898  esplyfv1  33903  mblfinlem2  38196  ftc1anclem8  38238  ftc1anc  38239  rhmpsr  43206  fvifeq  47905