Theorem eqeqan12d 2245

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
eqeqan12d.1	|- ( ph -> A = B )
eqeqan12d.2	|- ( ps -> C = D )

Assertion

Ref	Expression
eqeqan12d	|- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )

Proof of Theorem eqeqan12d

Step	Hyp	Ref	Expression
1		eqeqan12d.1	. 2 |- ( ph -> A = B )
2		eqeqan12d.2	. 2 |- ( ps -> C = D )
3		eqeq12 2242	. 2 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )
4	1, 2, 3	syl2an 289	1 |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395

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-5 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  eqeqan12rd  2246  eqfnfv  5731  eqfnfv2  5732  f1mpt  5894  xpopth  6320  f1o2ndf1  6372  ecopoveq  6775  xpdom2  6986  djune  7241  addpipqqs  7553  enq0enq  7614  enq0sym  7615  enq0tr  7617  enq0breq  7619  preqlu  7655  cnegexlem1  8317  neg11  8393  subeqrev  8518  cnref1o  9842  xneg11  10026  modlteq  10614  sq11  10829  qsqeqor  10867  fz1eqb  11007  eqwrd  11107  s111  11159  ccatopth  11243  wrd2ind  11250  cj11  11411  sqrt11  11545  sqabs  11588  recan  11615  reeff1  12206  efieq  12241  xpsff1o  13377  ismhm  13489  isdomn  14227  tgtop11  14744  ioocosf1o  15522  mpodvdsmulf1o  15658  iswlk  16029