Theorem ifeq12d 3581

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 3579	. 2 |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )
3		ifeq12d.2	. . 3 |- ( ph -> C = D )
4	3	ifeq2d 3580	. 2 |- ( ph -> if ( ps , B , C ) = if ( ps , B , D ) )
5	2, 4	eqtrd 2229	1 |- ( ph -> if ( ps , A , C ) = if ( ps , B , D ) )

Syntax hints:   -> wi 4   = wceq 1364   ifcif 3562

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-un 3161  df-if 3563

This theorem is referenced by:  ifbieq12d  3588  xaddpnf1  9938  exp3val  10650  eucalgval  12247  ennnfonelemp1  12648  ennnfonelemnn0  12664  mulgfvalg  13327  mulgpropdg  13370  lgsval  15329