Theorem ifeq12d 3642

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 3640	. 2 |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )
3		ifeq12d.2	. . 3 |- ( ph -> C = D )
4	3	ifeq2d 3641	. 2 |- ( ph -> if ( ps , B , C ) = if ( ps , B , D ) )
5	2, 4	eqtrd 2265	1 |- ( ph -> if ( ps , A , C ) = if ( ps , B , D ) )

Syntax hints:   -> wi 4   = wceq 1398   ifcif 3620

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2815  df-un 3215  df-if 3621

This theorem is referenced by:  ifbieq12d  3649  xaddpnf1  10179  exp3val  10903  swrdccat3blem  11431  eucalgval  12751  ennnfonelemp1  13157  ennnfonelemnn0  13173  mulgfvalg  13838  mulgpropdg  13881  lgsval  15877