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Mirrors > Home > MPE Home > Th. List > syl213anc | Structured version Visualization version GIF version |
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) |
Ref | Expression |
---|---|
syl12anc.1 | ⊢ (𝜑 → 𝜓) |
syl12anc.2 | ⊢ (𝜑 → 𝜒) |
syl12anc.3 | ⊢ (𝜑 → 𝜃) |
syl22anc.4 | ⊢ (𝜑 → 𝜏) |
syl23anc.5 | ⊢ (𝜑 → 𝜂) |
syl33anc.6 | ⊢ (𝜑 → 𝜁) |
syl213anc.7 | ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃 ∧ (𝜏 ∧ 𝜂 ∧ 𝜁)) → 𝜎) |
Ref | Expression |
---|---|
syl213anc | ⊢ (𝜑 → 𝜎) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl12anc.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | syl12anc.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | 1, 2 | jca 554 | . 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒)) |
4 | syl12anc.3 | . 2 ⊢ (𝜑 → 𝜃) | |
5 | syl22anc.4 | . 2 ⊢ (𝜑 → 𝜏) | |
6 | syl23anc.5 | . 2 ⊢ (𝜑 → 𝜂) | |
7 | syl33anc.6 | . 2 ⊢ (𝜑 → 𝜁) | |
8 | syl213anc.7 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃 ∧ (𝜏 ∧ 𝜂 ∧ 𝜁)) → 𝜎) | |
9 | 3, 4, 5, 6, 7, 8 | syl113anc 1338 | 1 ⊢ (𝜑 → 𝜎) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 197 df-an 386 df-3an 1039 |
This theorem is referenced by: syl223anc 1352 decpmatmul 20577 nosupbnd1lem5 31858 |
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