Theorem eleq12 2303

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 2301	. 2 |- ( A = B -> ( A e. C <-> B e. C ) )
2		eleq2 2302	. 2 |- ( C = D -> ( B e. C <-> B e. D ) )
3	1, 2	sylan9bb 466	1 |- ( ( A = B /\ C = D ) -> ( A e. C <-> B e. D ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209

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-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is used by:  trel  4236  pwnss  4296  epelg  4435  preleq  4702  acexmid  6084  cldval  15200