Theorem eleq12 2261

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 2259	. 2 |- ( A = B -> ( A e. C <-> B e. C ) )
2		eleq2 2260	. 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 1364   e. wcel 2167

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  trel  4139  pwnss  4193  epelg  4326  preleq  4592  acexmid  5924  cldval  14419