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Theorem ifov 7248
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 7156 . 2 (if(𝜑, 𝐹, 𝐺) = 𝐹 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐹𝐵))
2 oveq 7156 . 2 (if(𝜑, 𝐹, 𝐺) = 𝐺 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐺𝐵))
31, 2ifsb 4480 1 (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  ifcif 4467  (class class class)co 7150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-rex 3144  df-if 4468  df-uni 4833  df-br 5060  df-iota 6309  df-fv 6358  df-ov 7153
This theorem is referenced by:  monmatcollpw  21381
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