This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem eleq12i

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

Ref Expression
Hypotheses eleq1i.1 ⊢ 𝐴 = 𝐵
eleq12i.2 ⊢ 𝐶 = 𝐷
Assertion eleq12i ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷 )

Proof

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