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| Mirrors > Home > MPE Home > Th. List > sylnibr | Structured version Visualization version GIF version | ||
| Description: A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.) | 
| Ref | Expression | 
|---|---|
| sylnibr.1 | ⊢ (𝜑 → ¬ 𝜓) | 
| sylnibr.2 | ⊢ (𝜒 ↔ 𝜓) | 
| Ref | Expression | 
|---|---|
| sylnibr | ⊢ (𝜑 → ¬ 𝜒) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sylnibr.1 | . 2 ⊢ (𝜑 → ¬ 𝜓) | |
| 2 | sylnibr.2 | . . 3 ⊢ (𝜒 ↔ 𝜓) | |
| 3 | 2 | bicomi 224 | . 2 ⊢ (𝜓 ↔ 𝜒) | 
| 4 | 1, 3 | sylnib 328 | 1 ⊢ (𝜑 → ¬ 𝜒) | 
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