Theorem ifeq2 3946

Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
ifeq2

Proof of Theorem ifeq2

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabeq 3103	. . 3
2	1	uneq2d 3657	. 2
3		dfif6 3944	. 2
4		dfif6 3944	. 2
5	2, 3, 4	3eqtr4g 2523	1

Syntax hints:  -.wn 3  ->wi 4  =wceq 1395  {crab 2811  u.cun 3473  ifcif 3941

This theorem is referenced by:  ifeq12  3958  ifeq2d  3960  ifbieq2i  3965  ifexg  4011  somincom  5409  mdetunilem9  19122  prmorcht  23452  pclogsum  23490  ftc1anclem6  30095  ftc1anclem8  30097  ftc1anc  30098  hdmap1cbv  37530

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-nfc 2607  df-rab 2816  df-v 3111  df-un 3480  df-if 3942