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Theorem ifsb 3954
 Description: Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)
Hypotheses
Ref Expression
ifsb.1
ifsb.2
Assertion
Ref Expression
ifsb

Proof of Theorem ifsb
StepHypRef Expression
1 iftrue 3947 . . . 4
2 ifsb.1 . . . 4
31, 2syl 16 . . 3
4 iftrue 3947 . . 3
53, 4eqtr4d 2501 . 2
6 iffalse 3950 . . . 4
7 ifsb.2 . . . 4
86, 7syl 16 . . 3
9 iffalse 3950 . . 3
108, 9eqtr4d 2501 . 2
115, 10pm2.61i 164 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  =wceq 1395  ifcif 3941 This theorem is referenced by:  fvif  5882  iffv  5883  ovif  6379  ovif2  6380  ifov  6382  xmulneg1  11490  efrlim  23299  lgsneg  23594  lgsdilem  23597  rpvmasum2  23697 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-if 3942
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