Theorem ifeq12d 3565

Description: Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)

Hypotheses

Ref	Expression
ifeq1d.1	|- ( ph -> A = B )
ifeq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
ifeq12d	|- ( ph -> if ( ps , A , C ) = if ( ps , B , D ) )

Proof of Theorem ifeq12d

Step	Hyp	Ref	Expression
1		ifeq1d.1	. . 3 |- ( ph -> A = B )
2	1	ifeq1d 3563	. 2 |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )
3		ifeq12d.2	. . 3 |- ( ph -> C = D )
4	3	ifeq2d 3564	. 2 |- ( ph -> if ( ps , B , C ) = if ( ps , B , D ) )
5	2, 4	eqtrd 2220	1 |- ( ph -> if ( ps , A , C ) = if ( ps , B , D ) )

Syntax hints:   -> wi 4   = wceq 1363   ifcif 3546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-v 2751  df-un 3145  df-if 3547

This theorem is referenced by:  ifbieq12d  3572  xaddpnf1  9859  exp3val  10535  eucalgval  12067  ennnfonelemp1  12420  ennnfonelemnn0  12436  mulgfvalg  13013  mulgpropdg  13054  lgsval  14632