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Mirrors > Home > ILE Home > Th. List > mp2d | GIF version |
Description: A double modus ponens deduction. (Contributed by NM, 23-May-2013.) (Proof shortened by Wolf Lammen, 23-Jul-2013.) |
Ref | Expression |
---|---|
mp2d.1 | ⊢ (𝜑 → 𝜓) |
mp2d.2 | ⊢ (𝜑 → 𝜒) |
mp2d.3 | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) |
Ref | Expression |
---|---|
mp2d | ⊢ (𝜑 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mp2d.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | mp2d.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | mp2d.3 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) | |
4 | 2, 3 | mpid 42 | . 2 ⊢ (𝜑 → (𝜓 → 𝜃)) |
5 | 1, 4 | mpd 13 | 1 ⊢ (𝜑 → 𝜃) |
Colors of variables: wff set 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: fisseneq 6906 prloc 7442 axcaucvglemres 7850 bezoutlemmain 11942 coprm 12087 sqrt2irr 12105 oddprmdvds 12295 xblss2ps 13159 xblss2 13160 lgsprme0 13698 pw1nct 13998 apdiff 14042 |
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