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  5740  eqfnfv2  5741  f1mpt  5907  xpopth  6334  f1o2ndf1  6388  ecopoveq  6794  xpdom2  7010  djune  7268  addpipqqs  7580  enq0enq  7641  enq0sym  7642  enq0tr  7644  enq0breq  7646  preqlu  7682  cnegexlem1  8344  neg11  8420  subeqrev  8545  cnref1o  9875  xneg11  10059  modlteq  10649  sq11  10864  qsqeqor  10902  fz1eqb  11042  eqwrd  11144  s111  11198  ccatopth  11287  wrd2ind  11294  cj11  11456  sqrt11  11590  sqabs  11633  recan  11660  reeff1  12251  efieq  12286  xpsff1o  13422  ismhm  13534  isdomn  14273  tgtop11  14790  ioocosf1o  15568  mpodvdsmulf1o  15704  iswlk  16120