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Theorem ifsb 4049
 Description: Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)
Hypotheses
Ref Expression
ifsb.1 (if(𝜑, 𝐴, 𝐵) = 𝐴𝐶 = 𝐷)
ifsb.2 (if(𝜑, 𝐴, 𝐵) = 𝐵𝐶 = 𝐸)
Assertion
Ref Expression
ifsb 𝐶 = if(𝜑, 𝐷, 𝐸)

Proof of Theorem ifsb
StepHypRef Expression
1 iftrue 4042 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2 ifsb.1 . . . 4 (if(𝜑, 𝐴, 𝐵) = 𝐴𝐶 = 𝐷)
31, 2syl 17 . . 3 (𝜑𝐶 = 𝐷)
4 iftrue 4042 . . 3 (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷)
53, 4eqtr4d 2647 . 2 (𝜑𝐶 = if(𝜑, 𝐷, 𝐸))
6 iffalse 4045 . . . 4 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7 ifsb.2 . . . 4 (if(𝜑, 𝐴, 𝐵) = 𝐵𝐶 = 𝐸)
86, 7syl 17 . . 3 𝜑𝐶 = 𝐸)
9 iffalse 4045 . . 3 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸)
108, 9eqtr4d 2647 . 2 𝜑𝐶 = if(𝜑, 𝐷, 𝐸))
115, 10pm2.61i 175 1 𝐶 = if(𝜑, 𝐷, 𝐸)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475  ifcif 4036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-if 4037 This theorem is referenced by:  fvif  6114  iffv  6115  ovif  6635  ovif2  6636  ifov  6638  xmulneg1  11971  efrlim  24496  lgsneg  24846  lgsdilem  24849  rpvmasum2  25001
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