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Mirrors > Home > ILE Home > Th. List > mpdi | GIF version |
Description: A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by O'Cat, 15-Jan-2008.) |
Ref | Expression |
---|---|
mpdi.1 | ⊢ (𝜓 → 𝜒) |
mpdi.2 | ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) |
Ref | Expression |
---|---|
mpdi | ⊢ (𝜑 → (𝜓 → 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpdi.1 | . . 3 ⊢ (𝜓 → 𝜒) | |
2 | 1 | a1i 9 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | mpdi.2 | . 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) | |
4 | 2, 3 | mpdd 41 | 1 ⊢ (𝜑 → (𝜓 → 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: mpii 44 pm2.43d 50 gencbval 2778 sbcimdv 3020 suctr 4404 tfrlem9 6296 lbzbi 9564 flqeqceilz 10263 ndvdsadd 11879 gcdneg 11926 bj-inf2vnlem2 13968 |
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