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Theorem syl331anc 1501
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 1122 . 2 (𝜑 → (𝜏𝜂𝜁))
8 syl133anc.7 . 2 (𝜑𝜎)
9 syl331anc.8 . 2 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ 𝜎) → 𝜌)
101, 2, 3, 7, 8, 9syl311anc 1490 1 (𝜑𝜌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071
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 383  df-3an 1073
This theorem is referenced by:  syl332anc  1507  syl333anc  1508  qredeu  15579  brbtwn2  26006  3atlem4  35294  3atlem6  35296  llnexchb2  35677  osumcllem9N  35772  cdlemd4  36010  cdleme26fALTN  36171  cdleme26f  36172  cdleme36m  36270  cdlemg17b  36471  cdlemg17h  36477  cdlemk38  36724  cdlemk53b  36765  cdlemkyyN  36771  cdlemk43N  36772
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