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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 512 | . 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 1393 | 1 ⊢ (𝜑 → 𝜇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 |
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 206 df-an 397 df-3an 1088 |
This theorem is referenced by: 4atlem11 37623 dalem52 37738 dath2 37751 dalawlem1 37885 dalaw 37900 cdlemb2 38055 4atexlem7 38089 cdleme7ga 38262 cdleme18a 38305 cdleme18c 38307 cdleme21f 38346 cdleme26f2ALTN 38378 cdleme26f2 38379 cdleme27a 38381 cdlemg17dN 38677 cdlemg18a 38692 cdlemg31d 38714 cdlemg48 38751 cdlemj1 38835 dihord4 39272 |
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