Theorem ovif12 7458

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 4485	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2		iftrue 4485	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐶)
3	1, 2	oveq12d 7376	. . 3 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐴𝐹𝐶))
4		iftrue 4485	. . 3 ⊢ (𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐴𝐹𝐶))
5	3, 4	eqtr4d 2774	. 2 ⊢ (𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)))
6		iffalse 4488	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7		iffalse 4488	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐶, 𝐷) = 𝐷)
8	6, 7	oveq12d 7376	. . 3 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = (𝐵𝐹𝐷))
9		iffalse 4488	. . 3 ⊢ (¬ 𝜑 → if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) = (𝐵𝐹𝐷))
10	8, 9	eqtr4d 2774	. 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 1541  ifcif 4479  (class class class)co 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361

This theorem is referenced by:  ofccat  14892  limccnp2  25849  esplyind  33731  ftc1anclem5  37898  fsuppind  42843  sqrtcval  43892