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Mirrors > Home > MPE Home > Th. List > syl323anc | 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) |
syl23anc.5 | ⊢ (𝜑 → 𝜂) |
syl33anc.6 | ⊢ (𝜑 → 𝜁) |
syl133anc.7 | ⊢ (𝜑 → 𝜎) |
syl233anc.8 | ⊢ (𝜑 → 𝜌) |
syl323anc.9 | ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎 ∧ 𝜌)) → 𝜇) |
Ref | Expression |
---|---|
syl323anc | ⊢ (𝜑 → 𝜇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3anc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | syl3anc.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | syl3anc.3 | . 2 ⊢ (𝜑 → 𝜃) | |
4 | syl3Xanc.4 | . . 3 ⊢ (𝜑 → 𝜏) | |
5 | syl23anc.5 | . . 3 ⊢ (𝜑 → 𝜂) | |
6 | 4, 5 | jca 514 | . 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂)) |
7 | syl33anc.6 | . 2 ⊢ (𝜑 → 𝜁) | |
8 | syl133anc.7 | . 2 ⊢ (𝜑 → 𝜎) | |
9 | syl233anc.8 | . 2 ⊢ (𝜑 → 𝜌) | |
10 | syl323anc.9 | . 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎 ∧ 𝜌)) → 𝜇) | |
11 | 1, 2, 3, 6, 7, 8, 9, 10 | syl313anc 1390 | 1 ⊢ (𝜑 → 𝜇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 |
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 209 df-an 399 df-3an 1085 |
This theorem is referenced by: 4atlem11 36760 dalem52 36875 dath2 36888 dalawlem1 37022 dalaw 37037 cdlemb2 37192 4atexlem7 37226 cdleme7ga 37399 cdleme18a 37442 cdleme18c 37444 cdleme21f 37483 cdleme26f2ALTN 37515 cdleme26f2 37516 cdleme27a 37518 cdlemg17dN 37814 cdlemg18a 37829 cdlemg31d 37851 cdlemg48 37888 cdlemj1 37972 dihord4 38409 |
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