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

Step	Hyp	Ref	Expression
1		iftrue 3947	. . . 4
2		ifsb.1	. . . 4
3	1, 2	syl 16	. . 3
4		iftrue 3947	. . 3
5	3, 4	eqtr4d 2501	. 2
6		iffalse 3950	. . . 4
7		ifsb.2	. . . 4
8	6, 7	syl 16	. . 3
9		iffalse 3950	. . 3
10	8, 9	eqtr4d 2501	. 2
11	5, 10	pm2.61i 164	1

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