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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 323 | . 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)) |
3 | sylanl1.2 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) | |
4 | 2, 3 | sylan 267 | 1 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
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: adantlll 449 adantllr 450 isocnv 5451 nqnq0pi 6536 nqpnq0nq 6551 addnqprl 6627 addnqpru 6628 |
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