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Mirrors > Home > NFE 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 10 | . 2 ⊢ (φ → (ψ ↔ χ)) |
4 | 1, 3 | mtbid 291 | 1 ⊢ (φ → ¬ χ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 177 |
This theorem is referenced by: sylnibr 296 ssnelpss 3613 nnc3n3p1 6278 nchoicelem1 6289 |
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