Theorem eleq12i 1154

Description: Inference from equality to equivalence of membership.

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 1153	. 2 ⊢ (A ∈ C ↔ A ∈ D)
3		eleq1i.1	. . 3 ⊢ A = B
4	3	eleq1i 1152	. 2 ⊢ (A ∈ D ↔ B ∈ D)
5	2, 4	bitr 151	1 ⊢ (A ∈ C ↔ B ∈ D)

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

This theorem is referenced by:  1q 3851  0r 3983  1r 3984  m1r 3985

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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