Theorem eleq12 2144

Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)

Assertion

Ref	Expression
eleq12	|- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) )

Proof of Theorem eleq12

Step	Hyp	Ref	Expression
1		eleq1 2142	. 2 |- ( A = B -> ( A e. C <-> B e. C ) )
2		eleq2 2143	. 2 |- ( C = D -> ( B e. C <-> B e. D ) )
3	1, 2	sylan9bb 450	1 |- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078

This theorem is referenced by:  trel  3884  pwnss  3935  epelg  4047  preleq  4300  acexmid  5536