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Mirrors > Home > MPE Home > Th. List > syl112anc | Structured version Visualization version GIF version |
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) |
Ref | Expression |
---|---|
syl3anc.1 | ⊢ (𝜑 → 𝜓) |
syl3anc.2 | ⊢ (𝜑 → 𝜒) |
syl3anc.3 | ⊢ (𝜑 → 𝜃) |
syl3Xanc.4 | ⊢ (𝜑 → 𝜏) |
syl112anc.5 | ⊢ ((𝜓 ∧ 𝜒 ∧ (𝜃 ∧ 𝜏)) → 𝜂) |
Ref | Expression |
---|---|
syl112anc | ⊢ (𝜑 → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3anc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | syl3anc.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | syl3anc.3 | . . 3 ⊢ (𝜑 → 𝜃) | |
4 | syl3Xanc.4 | . . 3 ⊢ (𝜑 → 𝜏) | |
5 | 3, 4 | jca 515 | . 2 ⊢ (𝜑 → (𝜃 ∧ 𝜏)) |
6 | syl112anc.5 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ (𝜃 ∧ 𝜏)) → 𝜂) | |
7 | 1, 2, 5, 6 | syl3anc 1373 | 1 ⊢ (𝜑 → 𝜂) |
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