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Mirrors > Home > ILE Home > Th. List > eqtrd | GIF version |
Description: An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
eqtrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqtrd.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
eqtrd | ⊢ (𝜑 → 𝐴 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqtrd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | eqtrd.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
3 | 2 | eqeq2d 2177 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐴 = 𝐶)) |
4 | 1, 3 | mpbid 146 | 1 ⊢ (𝜑 → 𝐴 = 𝐶) |
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