Theorem ifeq12d 3660

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 3658	. 2 |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )
3		ifeq12d.2	. . 3 |- ( ph -> C = D )
4	3	ifeq2d 3659	. 2 |- ( ph -> if ( ps , B , C ) = if ( ps , B , D ) )
5	2, 4	eqtrd 2271	1 |- ( ph -> if ( ps , A , C ) = if ( ps , B , D ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   ifcif 3638

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-un 3224  df-if 3639

This theorem is used by:  ifbieq12d  3667  xaddpnf1  10248  exp3val  10978  swrdccat3blem  11511  eucalgval  12832  ennnfonelemp1  13297  ennnfonelemnn0  13313  mulgfvalg  13924  mulgpropdg  13967  lgsval  16123