MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ifsb Structured version   Visualization version   GIF version

Theorem ifsb 4299
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 4292 . . . 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 4292 . . 3 (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷)
53, 4eqtr4d 2850 . 2 (𝜑𝐶 = if(𝜑, 𝐷, 𝐸))
6 iffalse 4295 . . . 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 4295 . . 3 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸)
108, 9eqtr4d 2850 . 2 𝜑𝐶 = if(𝜑, 𝐷, 𝐸))
115, 10pm2.61i 176 1 𝐶 = if(𝜑, 𝐷, 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1637  ifcif 4286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-if 4287
This theorem is referenced by:  fvif  6427  iffv  6428  ovif  6970  ovif2  6971  ifov  6973  xmulneg1  12320  efrlim  24916  lgsneg  25266  lgsdilem  25269  rpvmasum2  25421
  Copyright terms: Public domain W3C validator