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Mirrors > Home > MPE Home > Th. List > syl123anc | 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) |
syl123anc.7 | ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁)) → 𝜎) |
Ref | Expression |
---|---|
syl123anc | ⊢ (𝜑 → 𝜎) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3anc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | syl3anc.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | syl3anc.3 | . . 3 ⊢ (𝜑 → 𝜃) | |
4 | 2, 3 | jca 514 | . 2 ⊢ (𝜑 → (𝜒 ∧ 𝜃)) |
5 | syl3Xanc.4 | . 2 ⊢ (𝜑 → 𝜏) | |
6 | syl23anc.5 | . 2 ⊢ (𝜑 → 𝜂) | |
7 | syl33anc.6 | . 2 ⊢ (𝜑 → 𝜁) | |
8 | syl123anc.7 | . 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁)) → 𝜎) | |
9 | 1, 4, 5, 6, 7, 8 | syl113anc 1378 | 1 ⊢ (𝜑 → 𝜎) |
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: dvfsumlem2 24627 atbtwnexOLDN 36587 atbtwnex 36588 osumcllem7N 37102 lhpmcvr5N 37167 cdleme22f2 37487 cdlemefs32sn1aw 37554 cdlemg7aN 37765 cdlemg7N 37766 cdlemg8c 37769 cdlemg8 37771 cdlemg11aq 37778 cdlemg12b 37784 cdlemg12e 37787 cdlemg12g 37789 cdlemg13a 37791 cdlemg15a 37795 cdlemg17e 37805 cdlemg18d 37821 cdlemg19a 37823 cdlemg20 37825 cdlemg22 37827 cdlemg28a 37833 cdlemg29 37845 cdlemg44a 37871 cdlemk34 38050 cdlemn11pre 38350 dihord10 38363 dihord2pre 38365 dihmeetlem17N 38463 |
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