Theorem eleq12 2415

Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)

Assertion

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

Proof of Theorem eleq12

Step	Hyp	Ref	Expression
1		eleq1 2413	. 2 ⊢ (A = B → (A ∈ C ↔ B ∈ C))
2		eleq2 2414	. 2 ⊢ (C = D → (B ∈ C ↔ B ∈ D))
3	1, 2	sylan9bb 680	1 ⊢ ((A = B ∧ C = D) → (A ∈ C ↔ B ∈ D))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710

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

This theorem is used by:  nnsucelr  4429  ncfinlower  4484  sfindbl  4531