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Mirrors > Home > ILE Home > Th. List > bitri | GIF version |
Description: An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) |
Ref | Expression |
---|---|
bitri.1 | ⊢ (𝜑 ↔ 𝜓) |
bitri.2 | ⊢ (𝜓 ↔ 𝜒) |
Ref | Expression |
---|---|
bitri | ⊢ (𝜑 ↔ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bitri.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | biimpi 113 | . . 3 ⊢ (𝜑 → 𝜓) |
3 | bitri.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
4 | 2, 3 | sylib 127 | . 2 ⊢ (𝜑 → 𝜒) |
5 | 3 | biimpri 124 | . . 3 ⊢ (𝜒 → 𝜓) |
6 | 5, 1 | sylibr 137 | . 2 ⊢ (𝜒 → 𝜑) |
7 | 4, 6 | impbii 117 | 1 ⊢ (𝜑 ↔ 𝜒) |
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