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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 309 | . 2 ⊢ (𝜑 → (𝜑 ∧ 𝜓)) |
3 | mpanl2.2 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) | |
4 | 2, 3 | sylan 278 | 1 ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem is referenced by: mpanr1 429 mp3an2 1268 reuss 3296 tfri3 6170 prarloclemarch2 7075 prarloclemlt 7149 prsradd 7428 pitonnlem2 7481 axcnre 7513 muleqadd 8234 divdivap2 8288 addltmul 8750 |
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