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Mirrors > Home > MPE Home > Th. List > 3imtr4i | Structured version Visualization version GIF version |
Description: A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 3-Jan-1993.) |
Ref | Expression |
---|---|
3imtr4.1 | ⊢ (𝜑 → 𝜓) |
3imtr4.2 | ⊢ (𝜒 ↔ 𝜑) |
3imtr4.3 | ⊢ (𝜃 ↔ 𝜓) |
Ref | Expression |
---|---|
3imtr4i | ⊢ (𝜒 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3imtr4.2 | . . 3 ⊢ (𝜒 ↔ 𝜑) | |
2 | 3imtr4.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
3 | 1, 2 | sylbi 216 | . 2 ⊢ (𝜒 → 𝜓) |
4 | 3imtr4.3 | . 2 ⊢ (𝜃 ↔ 𝜓) | |
5 | 3, 4 | sylibr 233 | 1 ⊢ (𝜒 → 𝜃) |
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