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  5734  eqfnfv2  5735  f1mpt  5901  xpopth  6328  f1o2ndf1  6380  ecopoveq  6785  xpdom2  6998  djune  7256  addpipqqs  7568  enq0enq  7629  enq0sym  7630  enq0tr  7632  enq0breq  7634  preqlu  7670  cnegexlem1  8332  neg11  8408  subeqrev  8533  cnref1o  9858  xneg11  10042  modlteq  10631  sq11  10846  qsqeqor  10884  fz1eqb  11024  eqwrd  11125  s111  11179  ccatopth  11263  wrd2ind  11270  cj11  11431  sqrt11  11565  sqabs  11608  recan  11635  reeff1  12226  efieq  12261  xpsff1o  13397  ismhm  13509  isdomn  14248  tgtop11  14765  ioocosf1o  15543  mpodvdsmulf1o  15679  iswlk  16068