Theorem fvif 6877

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 6861	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐴))
2		fveq2 6861	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹‘𝐵))
3	1, 2	ifsb 4505	1 ⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵))

Syntax hints:   = wceq 1540  ifcif 4491  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522

This theorem is referenced by:  ccatco  14808  sumeq2ii  15666  prodeq2ii  15884  ruclem1  16206  xpsrnbas  17541  rhmmpl  22277  rhmply1vr1  22281  mat2pmat1  22626  decpmatid  22664  pmatcollpwscmatlem1  22683  copco  24925  pcopt  24929  pcopt2  24930  limccnp  25799  prmorcht  27095  pclogsum  27133  mblfinlem2  37659  ftc1anclem8  37701  ftc1anc  37702  rhmpsr  42547  fvifeq  47285