Theorem eqeqan12d 2221

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 2218	. 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 1373

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 1470  ax-gen 1472  ax-4 1533  ax-17 1549  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198

This theorem is referenced by:  eqeqan12rd  2222  eqfnfv  5677  eqfnfv2  5678  f1mpt  5840  xpopth  6262  f1o2ndf1  6314  ecopoveq  6717  xpdom2  6926  djune  7180  addpipqqs  7483  enq0enq  7544  enq0sym  7545  enq0tr  7547  enq0breq  7549  preqlu  7585  cnegexlem1  8247  neg11  8323  subeqrev  8448  cnref1o  9772  xneg11  9956  modlteq  10542  sq11  10757  qsqeqor  10795  fz1eqb  10935  eqwrd  11034  s111  11085  cj11  11216  sqrt11  11350  sqabs  11393  recan  11420  reeff1  12011  efieq  12046  xpsff1o  13181  ismhm  13293  isdomn  14031  tgtop11  14548  ioocosf1o  15326  mpodvdsmulf1o  15462