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Mirrors > Home > ILE Home > Th. List > sylanl1 | GIF version |
Description: A syllogism inference. (Contributed by NM, 10-Mar-2005.) |
Ref | Expression |
---|---|
sylanl1.1 | ⊢ (𝜑 → 𝜓) |
sylanl1.2 | ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
sylanl1 | ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylanl1.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | 1 | anim1i 338 | . 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)) |
3 | sylanl1.2 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) | |
4 | 2, 3 | sylan 281 | 1 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem is referenced by: adantlll 471 adantllr 472 adantl3r 503 isocnv 5712 mapxpen 6742 nqnq0pi 7246 nqpnq0nq 7261 addnqprl 7337 addnqpru 7338 setsn0fun 11996 |
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