Theorem ifsb 4099

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 4092	. . . 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 4092	. . 3 ⊢ (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷)
5	3, 4	eqtr4d 2659	. 2 ⊢ (𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))
6		iffalse 4095	. . . 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 4095	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸)
10	8, 9	eqtr4d 2659	. 2 ⊢ (¬ 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))
11	5, 10	pm2.61i 176	1 ⊢ 𝐶 = if(𝜑, 𝐷, 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1483  ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  fvif  6204  iffv  6205  ovif  6737  ovif2  6738  ifov  6740  xmulneg1  12099  efrlim  24696  lgsneg  25046  lgsdilem  25049  rpvmasum2  25201