Theorem fvif 6936

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 6920	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐴))
2		fveq2 6920	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐵))
3	1, 2	ifsb 4561	1 ⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵))

Syntax hints:   = wceq 1537  ifcif 4548  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581

This theorem is referenced by:  ccatco  14884  sumeq2ii  15741  prodeq2ii  15959  ruclem1  16279  xpsrnbas  17631  rhmmpl  22408  rhmply1vr1  22412  mat2pmat1  22759  decpmatid  22797  pmatcollpwscmatlem1  22816  copco  25070  pcopt  25074  pcopt2  25075  limccnp  25946  prmorcht  27239  pclogsum  27277  mblfinlem2  37618  ftc1anclem8  37660  ftc1anc  37661  rhmpsr  42507  fvifeq  47195