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Mirrors > Home > MPE Home > Th. List > mpanlr1 | Structured version Visualization version GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 30-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) |
Ref | Expression |
---|---|
mpanlr1.1 | ⊢ 𝜓 |
mpanlr1.2 | ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
mpanlr1 | ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpanlr1.1 | . . 3 ⊢ 𝜓 | |
2 | 1 | jctl 524 | . 2 ⊢ (𝜒 → (𝜓 ∧ 𝜒)) |
3 | mpanlr1.2 | . 2 ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜏) | |
4 | 2, 3 | sylanl2 678 | 1 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 397 |
This theorem is referenced by: oecl 8354 omass 8398 oen0 8404 oeordi 8405 oewordri 8410 oeworde 8411 |
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