Theorem eleq12d 1157

Description: Deduction from equality to equivalence of membership.

Hypotheses

Ref	Expression
eleq1d.1	⊢ (φ → A = B)
eleq12d.2	⊢ (φ → C = D)

Assertion

Ref	Expression
eleq12d	⊢ (φ → (A ∈ C ↔ B ∈ D))

Proof of Theorem eleq12d

Step	Hyp	Ref	Expression
1		eleq12d.2	. . 3 ⊢ (φ → C = D)
2	1	eleq2d 1156	. 2 ⊢ (φ → (A ∈ C ↔ A ∈ D))
3		eleq1d.1	. . 3 ⊢ (φ → A = B)
4	3	eleq1d 1155	. 2 ⊢ (φ → (A ∈ D ↔ B ∈ D))
5	2, 4	bitrd 406	1 ⊢ (φ → (A ∈ C ↔ B ∈ D))

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

This theorem is referenced by:  ru 1437  sucidg 2305  canth 2945  tz7.49 2997  nnaordr 3178  omsmolem 3195  aceq3lem 3555  aceq5 3563  ac6lem 3575  numthlem 3598  ltapi 3824  ltmpi 3825

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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