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Theorem ifsb 4454
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 4447 . . . 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 4447 . . 3 (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷)
53, 4eqtr4d 2858 . 2 (𝜑𝐶 = if(𝜑, 𝐷, 𝐸))
6 iffalse 4450 . . . 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 4450 . . 3 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸)
108, 9eqtr4d 2858 . 2 𝜑𝐶 = if(𝜑, 𝐷, 𝐸))
115, 10pm2.61i 184 1 𝐶 = if(𝜑, 𝐷, 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  ifcif 4441
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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-if 4442
This theorem is referenced by:  fvif  6660  iffv  6661  ovif  7226  ovif2  7227  ifov  7229  xmulneg1  12639  efrlim  25531  lgsneg  25881  lgsdilem  25884  rpvmasum2  26072
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