Theorem eleq12i 2418

Description: Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Hypotheses

Ref	Expression
eleq1i.1	⊢ A = B
eleq12i.2	⊢ C = D

Assertion

Ref	Expression
eleq12i	⊢ (A ∈ C ↔ B ∈ D)

Proof of Theorem eleq12i

Step	Hyp	Ref	Expression
1		eleq12i.2	. . 3 ⊢ C = D
2	1	eleq2i 2417	. 2 ⊢ (A ∈ C ↔ A ∈ D)
3		eleq1i.1	. . 3 ⊢ A = B
4	3	eleq1i 2416	. 2 ⊢ (A ∈ D ↔ B ∈ D)
5	2, 4	bitri 240	1 ⊢ (A ∈ C ↔ B ∈ D)

Syntax hints:   ↔ wb 176   = 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:  3eltr3g  2435  3eltr4g  2436  sbcel12g  3152