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Mirrors > Home > MPE Home > Th. List > breqtrd | Structured version Visualization version GIF version |
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
Ref | Expression |
---|---|
breqtrd.1 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
breqtrd.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
breqtrd | ⊢ (𝜑 → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breqtrd.1 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | breqtrd.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
3 | 2 | breq2d 5065 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶)) |
4 | 1, 3 | mpbid 235 | 1 ⊢ (𝜑 → 𝐴𝑅𝐶) |
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