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Mirrors > Home > MPE Home > Th. List > syl222anc | 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) |
syl222anc.7 | ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁)) → 𝜎) |
Ref | Expression |
---|---|
syl222anc | ⊢ (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 511 | . 2 ⊢ (𝜑 → (𝜂 ∧ 𝜁)) |
8 | syl222anc.7 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁)) → 𝜎) | |
9 | 1, 2, 3, 4, 7, 8 | syl221anc 1380 | 1 ⊢ (𝜑 → 𝜎) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ 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 396 df-3an 1088 |
This theorem is referenced by: 3anandis 1470 3anandirs 1471 omopth2 8588 omeu 8589 dfac12lem2 10143 xaddass2 13234 xpncan 13235 divdenle 16690 pockthlem 16843 znidomb 21337 tanord1 26283 ang180lem5 26555 isosctrlem3 26562 log2tlbnd 26687 basellem1 26822 perfectlem2 26970 bposlem6 27029 dchrisum0flblem2 27249 pntpbnd1a 27325 mulsproplem1 27812 axcontlem8 28497 xlt2addrd 32239 s2f1 32379 xrge0addass 32459 xrge0npcan 32463 submatminr1 33089 carsgclctunlem2 33617 4atexlemntlpq 39243 4atexlemnclw 39245 trlval2 39338 cdleme0moN 39400 cdleme16b 39454 cdleme16c 39455 cdleme16d 39456 cdleme16e 39457 cdleme17c 39463 cdlemeda 39473 cdleme20h 39491 cdleme20j 39493 cdleme20l2 39496 cdleme21c 39502 cdleme21ct 39504 cdleme22aa 39514 cdleme22cN 39517 cdleme22d 39518 cdleme22e 39519 cdleme22eALTN 39520 cdleme23b 39525 cdleme25a 39528 cdleme25dN 39531 cdleme27N 39544 cdleme28a 39545 cdleme28b 39546 cdleme29ex 39549 cdleme32c 39618 cdleme42k 39659 cdlemg2cex 39766 cdlemg2idN 39771 cdlemg31c 39874 cdlemk5a 40010 cdlemk5 40011 congmul 42009 jm2.25lem1 42040 jm2.26 42044 jm2.27a 42047 infleinflem1 44379 stoweidlem42 45057 |
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