Theorem eleq12 2415

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 2413	. 2 |- (A = B -> (A e. C <-> B e. C))
2		eleq2 2414	. 2 |- (C = D -> (B e. C <-> B e. D))
3	1, 2	sylan9bb 680	1 |- ((A = B /\ C = D) -> (A e. C <-> B e. D))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349

This theorem is referenced by:  nnsucelr  4428  ncfinlower  4483  sfindbl  4530