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| Mirrors > Home > MPE Home > Th. List > syl21anc | Structured version Visualization version GIF version | ||
| Description: Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.) |
| Ref | Expression |
|---|---|
| syl12anc.1 | ⊢ (𝜑 → 𝜓) |
| syl12anc.2 | ⊢ (𝜑 → 𝜒) |
| syl12anc.3 | ⊢ (𝜑 → 𝜃) |
| syl21anc.4 | ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
| Ref | Expression |
|---|---|
| syl21anc | ⊢ (𝜑 → 𝜏) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | syl12anc.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 2 | syl12anc.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
| 3 | 1, 2 | jca 511 | . 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒)) |
| 4 | syl12anc.3 | . 2 ⊢ (𝜑 → 𝜃) | |
| 5 | syl21anc.4 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) | |
| 6 | 3, 4, 5 | syl2anc 584 | 1 ⊢ (𝜑 → 𝜏) |
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