Theorem eleq12i 2102

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 2101	. 2 ⊢ (A ∈ 𝐶 ↔ A ∈ 𝐷)
3		eleq1i.1	. . 3 ⊢ A = B
4	3	eleq1i 2100	. 2 ⊢ (A ∈ 𝐷 ↔ B ∈ 𝐷)
5	2, 4	bitri 173	1 ⊢ (A ∈ 𝐶 ↔ B ∈ 𝐷)

Syntax hints:   ↔ wb 98   = wceq 1242   ∈ wcel 1390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033

This theorem is referenced by:  3eltr3g  2119  3eltr4g  2120  sbcel12g  2859