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Theorem syl331anc 1432
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
Hypotheses
Ref Expression
syl12anc.1 (𝜑𝜓)
syl12anc.2 (𝜑𝜒)
syl12anc.3 (𝜑𝜃)
syl22anc.4 (𝜑𝜏)
syl23anc.5 (𝜑𝜂)
syl33anc.6 (𝜑𝜁)
syl133anc.7 (𝜑𝜎)
syl331anc.8 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ 𝜎) → 𝜌)
Assertion
Ref Expression
syl331anc (𝜑𝜌)

Proof of Theorem syl331anc
StepHypRef Expression
1 syl12anc.1 . 2 (𝜑𝜓)
2 syl12anc.2 . 2 (𝜑𝜒)
3 syl12anc.3 . 2 (𝜑𝜃)
4 syl22anc.4 . . 3 (𝜑𝜏)
5 syl23anc.5 . . 3 (𝜑𝜂)
6 syl33anc.6 . . 3 (𝜑𝜁)
74, 5, 63jca 1315 . 2 (𝜑 → (𝜏𝜂𝜁))
8 syl133anc.7 . 2 (𝜑𝜎)
9 syl331anc.8 . 2 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ 𝜎) → 𝜌)
101, 2, 3, 7, 8, 9syl311anc 1421 1 (𝜑𝜌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072
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 1074
This theorem is referenced by:  syl332anc  1438  syl333anc  1439  qredeu  15463  brbtwn2  25873  3atlem4  35160  3atlem6  35162  llnexchb2  35543  osumcllem9N  35638  cdlemd4  35876  cdleme26fALTN  36037  cdleme26f  36038  cdleme36m  36136  cdlemg17b  36337  cdlemg17h  36343  cdlemk38  36590  cdlemk53b  36631  cdlemkyyN  36637  cdlemk43N  36638
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