![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > mpanl1 | GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) |
Ref | Expression |
---|---|
mpanl1.1 | ⊢ 𝜑 |
mpanl1.2 | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
mpanl1 | ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpanl1.1 | . . 3 ⊢ 𝜑 | |
2 | 1 | jctl 314 | . 2 ⊢ (𝜓 → (𝜑 ∧ 𝜓)) |
3 | mpanl1.2 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) | |
4 | 2, 3 | sylan 283 | 1 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem is referenced by: mpanl12 436 ercnv 6555 rec11api 8708 divdiv23apzi 8720 recp1lt1 8854 divgt0i 8865 divge0i 8866 ltreci 8867 lereci 8868 lt2msqi 8869 le2msqi 8870 msq11i 8871 ltdiv23i 8881 fnn0ind 9367 elfzp1b 10094 elfzm1b 10095 sqrt11i 11136 sqrtmuli 11137 sqrtmsq2i 11139 sqrtlei 11140 sqrtlti 11141 |
Copyright terms: Public domain | W3C validator |