Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ifov Structured version   Visualization version   GIF version

Theorem ifov 6737
 Description: Move a conditional outside of an operation. (Contributed by AV, 11-Nov-2019.)
Assertion
Ref Expression
ifov (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵))

Proof of Theorem ifov
StepHypRef Expression
1 oveq 6653 . 2 (if(𝜑, 𝐹, 𝐺) = 𝐹 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐹𝐵))
2 oveq 6653 . 2 (if(𝜑, 𝐹, 𝐺) = 𝐺 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐺𝐵))
31, 2ifsb 4097 1 (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1482  ifcif 4084  (class class class)co 6647 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rex 2917  df-if 4085  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650 This theorem is referenced by:  monmatcollpw  20578
 Copyright terms: Public domain W3C validator