Theorem eleq12 1959

Description: Equality implies equivalence of membership.

Assertion

Ref	Expression
eleq12	|- ((A = B /\ C = D) -> (A e. C <-> B e. D))

Proof of Theorem eleq12

Step	Hyp	Ref	Expression
1		eleq1 1957	. 2 |- (A = B -> (A e. C <-> B e. C))
2		eleq2 1958	. 2 |- (C = D -> (B e. C <-> B e. D))
3	1, 2	sylan9bb 599	1 |- ((A = B /\ C = D) -> (A e. C <-> B e. D))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300

This theorem is referenced by:  trel 3418  preleq 5708

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-cleq 1877  df-clel 1880

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