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Mirrors > Home > ILE Home > Th. List > breq2 | GIF version |
Description: Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.) |
Ref | Expression |
---|---|
breq2 | ⊢ (A = B → (𝐶𝑅A ↔ 𝐶𝑅B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 3541 | . . 3 ⊢ (A = B → 〈𝐶, A〉 = 〈𝐶, B〉) | |
2 | 1 | eleq1d 2103 | . 2 ⊢ (A = B → (〈𝐶, A〉 ∈ 𝑅 ↔ 〈𝐶, B〉 ∈ 𝑅)) |
3 | df-br 3756 | . 2 ⊢ (𝐶𝑅A ↔ 〈𝐶, A〉 ∈ 𝑅) | |
4 | df-br 3756 | . 2 ⊢ (𝐶𝑅B ↔ 〈𝐶, B〉 ∈ 𝑅) | |
5 | 2, 3, 4 | 3bitr4g 212 | 1 ⊢ (A = B → (𝐶𝑅A ↔ 𝐶𝑅B)) |
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