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Mirrors > Home > MPE Home > Th. List > syl312anc | Structured version Visualization version GIF version |
Description: Syllogism combined with contraction. (Contributed by NM, 11-Jul-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) |
syl312anc.7 | ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ (𝜂 ∧ 𝜁)) → 𝜎) |
Ref | Expression |
---|---|
syl312anc | ⊢ (𝜑 → 𝜎) |
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 | . 2 ⊢ (𝜑 → 𝜏) | |
5 | syl23anc.5 | . . 3 ⊢ (𝜑 → 𝜂) | |
6 | syl33anc.6 | . . 3 ⊢ (𝜑 → 𝜁) | |
7 | 5, 6 | jca 513 | . 2 ⊢ (𝜑 → (𝜂 ∧ 𝜁)) |
8 | syl312anc.7 | . 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ (𝜂 ∧ 𝜁)) → 𝜎) | |
9 | 1, 2, 3, 4, 7, 8 | syl311anc 1385 | 1 ⊢ (𝜑 → 𝜎) |
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: pythagtriplem19 16766 cdleme27cl 39237 cdlemefs27cl 39284 cdleme32fvcl 39311 cdlemg16ALTN 39529 cdlemg27a 39563 cdlemg31c 39570 cdlemg39 39587 cdlemk11ta 39800 cdlemk19ylem 39801 cdlemk11tc 39816 cdlemk45 39818 dihmeetlem12N 40189 dihjatc 40288 flt4lem5c 41396 flt4lem5d 41397 flt4lem5e 41398 |
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