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Mirrors > Home > ILE Home > Th. List > mpanr2 | GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) |
Ref | Expression |
---|---|
mpanr2.1 | ⊢ 𝜒 |
mpanr2.2 | ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) |
Ref | Expression |
---|---|
mpanr2 | ⊢ ((𝜑 ∧ 𝜓) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpanr2.1 | . . 3 ⊢ 𝜒 | |
2 | 1 | jctr 311 | . 2 ⊢ (𝜓 → (𝜓 ∧ 𝜒)) |
3 | mpanr2.2 | . 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) | |
4 | 2, 3 | sylan2 282 | 1 ⊢ ((𝜑 ∧ 𝜓) → 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: op1steq 6031 fpmg 6522 pmresg 6524 pm54.43 6996 prarloclemarch2 7175 prarloclemlt 7249 prsradd 7528 muleqadd 8342 divdivap1 8396 addltmul 8860 elfzp1b 9770 elfzm1b 9771 expp1zap 10235 expm1ap 10236 fiinbas 12059 opnneissb 12167 blssec 12427 |
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