Theorem eleq12 2102

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 2100	. 2 |- ( A = B -> ( A e. C <-> B e. C ) )
2		eleq2 2101	. 2 |- ( C = D -> ( B e. C <-> B e. D ) )
3	1, 2	sylan9bb 435	1 |- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036

This theorem is referenced by:  trel  3861  pwnss  3912  epelg  4027  preleq  4279  acexmid  5511