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Mirrors > Home > ILE Home > Th. List > eleq1 | GIF version |
Description: Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
eleq1 | ⊢ (A = B → (A ∈ 𝐶 ↔ B ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2046 | . . . 4 ⊢ (A = B → (x = A ↔ x = B)) | |
2 | 1 | anbi1d 438 | . . 3 ⊢ (A = B → ((x = A ∧ x ∈ 𝐶) ↔ (x = B ∧ x ∈ 𝐶))) |
3 | 2 | exbidv 1703 | . 2 ⊢ (A = B → (∃x(x = A ∧ x ∈ 𝐶) ↔ ∃x(x = B ∧ x ∈ 𝐶))) |
4 | df-clel 2033 | . 2 ⊢ (A ∈ 𝐶 ↔ ∃x(x = A ∧ x ∈ 𝐶)) | |
5 | df-clel 2033 | . 2 ⊢ (B ∈ 𝐶 ↔ ∃x(x = B ∧ x ∈ 𝐶)) | |
6 | 3, 4, 5 | 3bitr4g 212 | 1 ⊢ (A = B → (A ∈ 𝐶 ↔ B ∈ 𝐶)) |
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