Theorem eleq12 1151

Description: Equality implies equivalence of membership.

Assertion

Ref	Expression
eleq12	⊢ ((A = B ∧ C = D) → (A ∈ C ↔ B ∈ D))

Proof of Theorem eleq12

Step	Hyp	Ref	Expression
1		eleq1 1149	. 2 ⊢ (A = B → (A ∈ C ↔ B ∈ C))
2		eleq2 1150	. 2 ⊢ (C = D → (B ∈ C ↔ B ∈ D))
3	1, 2	sylan9bb 418	1 ⊢ ((A = B ∧ C = D) → (A ∈ C ↔ B ∈ D))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  preleq 3454

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099

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