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Mirrors > Home > MPE Home > Th. List > syl322anc | Structured version Visualization version GIF version |
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) |
Ref | Expression |
---|---|
syl12anc.1 | ⊢ (𝜑 → 𝜓) |
syl12anc.2 | ⊢ (𝜑 → 𝜒) |
syl12anc.3 | ⊢ (𝜑 → 𝜃) |
syl22anc.4 | ⊢ (𝜑 → 𝜏) |
syl23anc.5 | ⊢ (𝜑 → 𝜂) |
syl33anc.6 | ⊢ (𝜑 → 𝜁) |
syl133anc.7 | ⊢ (𝜑 → 𝜎) |
syl322anc.8 | ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎)) → 𝜌) |
Ref | Expression |
---|---|
syl322anc | ⊢ (𝜑 → 𝜌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl12anc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | syl12anc.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | syl12anc.3 | . 2 ⊢ (𝜑 → 𝜃) | |
4 | syl22anc.4 | . 2 ⊢ (𝜑 → 𝜏) | |
5 | syl23anc.5 | . 2 ⊢ (𝜑 → 𝜂) | |
6 | syl33anc.6 | . . 3 ⊢ (𝜑 → 𝜁) | |
7 | syl133anc.7 | . . 3 ⊢ (𝜑 → 𝜎) | |
8 | 6, 7 | jca 553 | . 2 ⊢ (𝜑 → (𝜁 ∧ 𝜎)) |
9 | syl322anc.8 | . 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎)) → 𝜌) | |
10 | 1, 2, 3, 4, 5, 8, 9 | syl321anc 1388 | 1 ⊢ (𝜑 → 𝜌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 |
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 197 df-an 385 df-3an 1056 |
This theorem is referenced by: ax5seglem6 25859 ax5seg 25863 elpaddatriN 35407 paddasslem8 35431 paddasslem12 35435 paddasslem13 35436 pmodlem1 35450 osumcllem5N 35564 pexmidlem2N 35575 cdleme3h 35840 cdleme7ga 35853 cdleme20l 35927 cdleme21ct 35934 cdleme21d 35935 cdleme21e 35936 cdleme26e 35964 cdleme26eALTN 35966 cdleme26fALTN 35967 cdleme26f 35968 cdleme26f2ALTN 35969 cdleme26f2 35970 cdleme39n 36071 cdlemh2 36421 cdlemh 36422 cdlemk12 36455 cdlemk12u 36477 cdlemkfid1N 36526 congsub 37854 mzpcong 37856 jm2.18 37872 jm2.15nn0 37887 jm2.27c 37891 |
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