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Mirrors > Home > NFE Home > Th. List > mpsyl | GIF version |
Description: Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.) |
Ref | Expression |
---|---|
mpsyl.1 | ⊢ φ |
mpsyl.2 | ⊢ (ψ → χ) |
mpsyl.3 | ⊢ (φ → (χ → θ)) |
Ref | Expression |
---|---|
mpsyl | ⊢ (ψ → θ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpsyl.1 | . . 3 ⊢ φ | |
2 | 1 | a1i 10 | . 2 ⊢ (ψ → φ) |
3 | mpsyl.2 | . 2 ⊢ (ψ → χ) | |
4 | mpsyl.3 | . 2 ⊢ (φ → (χ → θ)) | |
5 | 2, 3, 4 | sylc 56 | 1 ⊢ (ψ → θ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 |
This theorem is referenced by: tbwlem1 1470 tbwlem2 1471 re1luk3 1477 merco1lem17 1498 re1tbw1 1510 a16g 1945 funmo 5125 foimacnv 5303 isoini2 5498 |
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