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Mirrors > Home > MPE Home > Th. List > 3brtr4d | Structured version Visualization version GIF version |
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.) |
Ref | Expression |
---|---|
3brtr4d.1 | ⊢ (𝜑 → 𝐴𝑅𝐵) |
3brtr4d.2 | ⊢ (𝜑 → 𝐶 = 𝐴) |
3brtr4d.3 | ⊢ (𝜑 → 𝐷 = 𝐵) |
Ref | Expression |
---|---|
3brtr4d | ⊢ (𝜑 → 𝐶𝑅𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3brtr4d.1 | . 2 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
2 | 3brtr4d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐴) | |
3 | 3brtr4d.3 | . . 3 ⊢ (𝜑 → 𝐷 = 𝐵) | |
4 | 2, 3 | breq12d 5087 | . 2 ⊢ (𝜑 → (𝐶𝑅𝐷 ↔ 𝐴𝑅𝐵)) |
5 | 1, 4 | mpbird 256 | 1 ⊢ (𝜑 → 𝐶𝑅𝐷) |
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