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

Theorem ifov 7375
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 7281 . 2 (if(𝜑, 𝐹, 𝐺) = 𝐹 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐹𝐵))
2 oveq 7281 . 2 (if(𝜑, 𝐹, 𝐺) = 𝐺 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐺𝐵))
31, 2ifsb 4472 1 (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  ifcif 4459  (class class class)co 7275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-if 4460  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278
This theorem is referenced by:  monmatcollpw  21928
  Copyright terms: Public domain W3C validator