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Mirrors > Home > ILE Home > Th. List > eleq2d | GIF version |
Description: Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.) |
Ref | Expression |
---|---|
eleq1d.1 | ⊢ (φ → A = B) |
Ref | Expression |
---|---|
eleq2d | ⊢ (φ → (𝐶 ∈ A ↔ 𝐶 ∈ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1d.1 | . 2 ⊢ (φ → A = B) | |
2 | eleq2 2098 | . 2 ⊢ (A = B → (𝐶 ∈ A ↔ 𝐶 ∈ B)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (φ → (𝐶 ∈ A ↔ 𝐶 ∈ B)) |
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