Theorem eleq1a 2422

Description: A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)

Assertion

Ref	Expression
eleq1a	⊢ (A ∈ B → (C = A → C ∈ B))

Proof of Theorem eleq1a

Step	Hyp	Ref	Expression
1		eleq1 2413	. 2 ⊢ (C = A → (C ∈ B ↔ A ∈ B))
2	1	biimprcd 216	1 ⊢ (A ∈ B → (C = A → C ∈ B))

Syntax hints:   → wi 4   = wceq 1642   ∈ 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:  elex22  2871  elex2  2872  reu6  3026  disjne  3597  evennn  4507  oddnn  4508  nnadjoin  4521  phi11lem1  4596  f1o2d  5728