Theorem ovif12 7533

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 4531	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2		iftrue 4531	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐶)
3	1, 2	oveq12d 7449	. . 3 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐴𝐹𝐶))
4		iftrue 4531	. . 3 ⊢ (𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐴𝐹𝐶))
5	3, 4	eqtr4d 2780	. 2 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)))
6		iffalse 4534	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7		iffalse 4534	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐷)
8	6, 7	oveq12d 7449	. . 3 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐵𝐹𝐷))
9		iffalse 4534	. . 3 ⊢ (¬ 𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐵𝐹𝐷))
10	8, 9	eqtr4d 2780	. 2 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)))
11	5, 10	pm2.61i 182	1 ⊢ (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷))

Syntax hints:  ¬ wn 3   = wceq 1540  ifcif 4525  (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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434

This theorem is referenced by:  ofccat  15008  limccnp2  25927  ftc1anclem5  37704  fsuppind  42600  sqrtcval  43654