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Mirrors > Home > ILE Home > Th. List > syland | GIF version |
Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.) |
Ref | Expression |
---|---|
syland.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
syland.2 | ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏)) |
Ref | Expression |
---|---|
syland | ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syland.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | syland.2 | . . . 4 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏)) | |
3 | 2 | expd 245 | . . 3 ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏))) |
4 | 1, 3 | syld 40 | . 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏))) |
5 | 4 | impd 242 | 1 ⊢ (𝜑 → ((𝜓 ∧ 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: sylan2d 278 syl2and 279 sylani 386 nn0seqcvgd 9880 |
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