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Mirrors > Home > ILE Home > Th. List > sylnib | GIF version |
Description: A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.) |
Ref | Expression |
---|---|
sylnib.1 | ⊢ (𝜑 → ¬ 𝜓) |
sylnib.2 | ⊢ (𝜓 ↔ 𝜒) |
Ref | Expression |
---|---|
sylnib | ⊢ (𝜑 → ¬ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylnib.1 | . 2 ⊢ (𝜑 → ¬ 𝜓) | |
2 | sylnib.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
3 | 2 | a1i 9 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
4 | 1, 3 | mtbid 597 | 1 ⊢ (𝜑 → ¬ 𝜒) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: sylnibr 602 ssnelpss 3289 inssdif0im 3291 ordtriexmidlem2 4246 dmsn0el 4790 fidifsnen 6331 ltpopr 6693 caucvgprprlemnbj 6791 xrlttri3 8718 fzneuz 8963 |
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