Theorem eleq12 1536

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 1534	. 2 |- (A = B -> (A e. C <-> B e. C))
2		eleq2 1535	. 2 |- (C = D -> (B e. C <-> B e. D))
3	1, 2	sylan9bb 540	1 |- ((A = B /\ C = D) -> (A e. C <-> B e. D))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958

This theorem is referenced by:  preleq 4603

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-cleq 1469  df-clel 1472

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