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| Mirrors > Home > MPE Home > Th. List > biimtrid | Structured version Visualization version GIF version | ||
| Description: A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 12-Jan-1993.) |
| Ref | Expression |
|---|---|
| biimtrid.1 | ⊢ (𝜑 ↔ 𝜓) |
| biimtrid.2 | ⊢ (𝜒 → (𝜓 → 𝜃)) |
| Ref | Expression |
|---|---|
| biimtrid | ⊢ (𝜒 → (𝜑 → 𝜃)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | biimtrid.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
| 2 | 1 | biimpi 216 | . 2 ⊢ (𝜑 → 𝜓) |
| 3 | biimtrid.2 | . 2 ⊢ (𝜒 → (𝜓 → 𝜃)) | |
| 4 | 2, 3 | syl5 34 | 1 ⊢ (𝜒 → (𝜑 → 𝜃)) |
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