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Mirrors > Home > ILE Home > Th. List > mpanl2 | GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
Ref | Expression |
---|---|
mpanl2.1 | ⊢ 𝜓 |
mpanl2.2 | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
mpanl2 | ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpanl2.1 | . . 3 ⊢ 𝜓 | |
2 | 1 | jctr 298 | . 2 ⊢ (𝜑 → (𝜑 ∧ 𝜓)) |
3 | mpanl2.2 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) | |
4 | 2, 3 | sylan 267 | 1 ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem is referenced by: mpanr1 413 mp3an2 1220 reuss 3218 tfri3 5953 prarloclemarch2 6517 prarloclemlt 6591 prsradd 6870 pitonnlem2 6923 axcnre 6955 muleqadd 7649 divdivap2 7700 addltmul 8161 |
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