Theorem ifsb 4488

Description: Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)

Hypotheses

Ref	Expression
ifsb.1	⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷)
ifsb.2	⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸)

Assertion

Ref	Expression
ifsb	⊢ 𝐶 = if(𝜑, 𝐷, 𝐸)

Proof of Theorem ifsb

Step	Hyp	Ref	Expression
1		iftrue 4480	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2		ifsb.1	. . . 4 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4		iftrue 4480	. . 3 ⊢ (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷)
5	3, 4	eqtr4d 2769	. 2 ⊢ (𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))
6		iffalse 4483	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7		ifsb.2	. . . 4 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸)
8	6, 7	syl 17	. . 3 ⊢ (¬ 𝜑 → 𝐶 = 𝐸)
9		iffalse 4483	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸)
10	8, 9	eqtr4d 2769	. 2 ⊢ (¬ 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))
11	5, 10	pm2.61i 182	1 ⊢ 𝐶 = if(𝜑, 𝐷, 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541  ifcif 4474

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 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-if 4475

This theorem is referenced by:  fvif  6844  iffv  6845  ovif  7450  ovif2  7451  ifov  7453  xmulneg1  13174  efrlim  26912  efrlimOLD  26913  lgsneg  27265  lgsdilem  27268  rpvmasum2  27456