Theorem ifov 7534

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

Step	Hyp	Ref	Expression
1		oveq 7437	. 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐹 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐹𝐵))
2		oveq 7437	. 2 ⊢ (if(𝜑, 𝐹, 𝐺) = 𝐺 → (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = (𝐴𝐺𝐵))
3	1, 2	ifsb 4544	1 ⊢ (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵))

Syntax hints:   = wceq 1537  ifcif 4531  (class class class)co 7431

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-if 4532  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434

This theorem is referenced by:  monmatcollpw  22801