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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 513 | . 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 1395 | 1 ⊢ (𝜑 → 𝜇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 |
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 398 df-3an 1090 |
This theorem is referenced by: 4atlem11 38122 dalem52 38237 dath2 38250 dalawlem1 38384 dalaw 38399 cdlemb2 38554 4atexlem7 38588 cdleme7ga 38761 cdleme18a 38804 cdleme18c 38806 cdleme21f 38845 cdleme26f2ALTN 38877 cdleme26f2 38878 cdleme27a 38880 cdlemg17dN 39176 cdlemg18a 39191 cdlemg31d 39213 cdlemg48 39250 cdlemj1 39334 dihord4 39771 |
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