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Mirrors > Home > NFE Home > Th. List > mp3an13 | GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
Ref | Expression |
---|---|
mp3an13.1 | ⊢ φ |
mp3an13.2 | ⊢ χ |
mp3an13.3 | ⊢ ((φ ∧ ψ ∧ χ) → θ) |
Ref | Expression |
---|---|
mp3an13 | ⊢ (ψ → θ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mp3an13.1 | . 2 ⊢ φ | |
2 | mp3an13.2 | . . 3 ⊢ χ | |
3 | mp3an13.3 | . . 3 ⊢ ((φ ∧ ψ ∧ χ) → θ) | |
4 | 2, 3 | mp3an3 1266 | . 2 ⊢ ((φ ∧ ψ) → θ) |
5 | 1, 4 | mpan 651 | 1 ⊢ (ψ → θ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 934 |
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 df-an 360 df-3an 936 |
This theorem is referenced by: sfinltfin 4535 nmembers1lem2 6269 |
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