Theorem eqeqan12d 2254

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 2251	. 2 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )
4	1, 2, 3	syl2an 289	1 |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231

This theorem is used by:  eqeqan12rd  2255  eqfnfv  5806  eqfnfv2  5807  f1mpt  5977  xpopth  6410  f1o2ndf1  6464  ecopoveq  6904  xpdom2  7129  djune  7418  addpipqqs  7737  enq0enq  7798  enq0sym  7799  enq0tr  7801  enq0breq  7803  preqlu  7839  cnegexlem1  8502  neg11  8578  subeqrev  8703  cnref1o  10061  xneg11  10246  modlteq  10847  sq11  11062  qsqeqor  11100  fz1eqb  11243  eqwrd  11359  s111  11413  ccatopth  11502  wrd2ind  11509  cj11  11685  sqrt11  11819  sqabs  11863  recan  11890  reeff1  12483  efieq  12518  xpsff1o  13719  ismhm  13817  isdomn  14627  tgtop11  15226  ioocosf1o  16005  mpodvdsmulf1o  16185  iswlk  16662