Theorem eleq12 2270

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 2268	. 2 |- ( A = B -> ( A e. C <-> B e. C ) )
2		eleq2 2269	. 2 |- ( C = D -> ( B e. C <-> B e. D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176

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-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201

This theorem is referenced by:  trel  4149  pwnss  4203  epelg  4337  preleq  4603  acexmid  5943  cldval  14571