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Mirrors > Home > ILE Home > Th. List > syl3anc | GIF version |
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) |
Ref | Expression |
---|---|
sylXanc.1 | ⊢ (𝜑 → 𝜓) |
sylXanc.2 | ⊢ (𝜑 → 𝜒) |
sylXanc.3 | ⊢ (𝜑 → 𝜃) |
syl111anc.4 | ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
syl3anc | ⊢ (𝜑 → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylXanc.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | sylXanc.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | sylXanc.3 | . . 3 ⊢ (𝜑 → 𝜃) | |
4 | 1, 2, 3 | 3jca 1166 | . 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) |
5 | syl111anc.4 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | |
6 | 4, 5 | syl 14 | 1 ⊢ (𝜑 → 𝜏) |
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