Theorem ovif12 7496

Description: Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 25-Jan-2017.)

Assertion

Ref	Expression
ovif12	⊢ (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷))

Proof of Theorem ovif12

Step	Hyp	Ref	Expression
1		iftrue 4486	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2		iftrue 4486	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐶)
3	1, 2	oveq12d 7414	. . 3 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐴𝐹𝐶))
4		iftrue 4486	. . 3 ⊢ (𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐴𝐹𝐶))
5	3, 4	eqtr4d 2800	. 2 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)))
6		iffalse 4489	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7		iffalse 4489	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐷)
8	6, 7	oveq12d 7414	. . 3 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐵𝐹𝐷))
9		iffalse 4489	. . 3 ⊢ (¬ 𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐵𝐹𝐷))
10	8, 9	eqtr4d 2800	. 2 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)))
11	5, 10	pm2.61i 183	1 ⊢ (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷))

Syntax hints:  ¬ wn 3   = wceq 1560  ifcif 4480  (class class class)co 7396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-ov 7399

This theorem is referenced by:  ofccat  14982  limccnp2  25954  esplyind  33872  ftc1anclem5  38196  fsuppind  43172  sqrtcval  44217