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Theorem syl233anc 1398
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
Hypotheses
Ref Expression
syl3anc.1 (𝜑𝜓)
syl3anc.2 (𝜑𝜒)
syl3anc.3 (𝜑𝜃)
syl3Xanc.4 (𝜑𝜏)
syl23anc.5 (𝜑𝜂)
syl33anc.6 (𝜑𝜁)
syl133anc.7 (𝜑𝜎)
syl233anc.8 (𝜑𝜌)
syl233anc.9 (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)
Assertion
Ref Expression
syl233anc (𝜑𝜇)

Proof of Theorem syl233anc
StepHypRef Expression
1 syl3anc.1 . . 3 (𝜑𝜓)
2 syl3anc.2 . . 3 (𝜑𝜒)
31, 2jca 511 . 2 (𝜑 → (𝜓𝜒))
4 syl3anc.3 . 2 (𝜑𝜃)
5 syl3Xanc.4 . 2 (𝜑𝜏)
6 syl23anc.5 . 2 (𝜑𝜂)
7 syl33anc.6 . 2 (𝜑𝜁)
8 syl133anc.7 . 2 (𝜑𝜎)
9 syl233anc.8 . 2 (𝜑𝜌)
10 syl233anc.9 . 2 (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)
113, 4, 5, 6, 7, 8, 9, 10syl133anc 1392 1 (𝜑𝜇)
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 207  df-an 396  df-3an 1088
This theorem is referenced by:  br8d  32630  2llnjN  39550  cdleme16b  40262  cdleme18d  40278  cdleme19d  40289  cdleme20bN  40293  cdleme20l1  40303  cdleme22cN  40325  cdleme22eALTN  40328  cdleme22f  40329  cdlemg33c0  40685  cdlemk5  40819  cdlemk5u  40844  cdlemky  40909  cdlemkyyN  40945
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