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Theorem syl321anc 1192
Description: Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
Hypotheses
Ref Expression
sylXanc.1 (𝜑𝜓)
sylXanc.2 (𝜑𝜒)
sylXanc.3 (𝜑𝜃)
sylXanc.4 (𝜑𝜏)
sylXanc.5 (𝜑𝜂)
sylXanc.6 (𝜑𝜁)
syl321anc.7 (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ 𝜁) → 𝜎)
Assertion
Ref Expression
syl321anc (𝜑𝜎)

Proof of Theorem syl321anc
StepHypRef Expression
1 sylXanc.1 . 2 (𝜑𝜓)
2 sylXanc.2 . 2 (𝜑𝜒)
3 sylXanc.3 . 2 (𝜑𝜃)
4 sylXanc.4 . . 3 (𝜑𝜏)
5 sylXanc.5 . . 3 (𝜑𝜂)
64, 5jca 300 . 2 (𝜑 → (𝜏𝜂))
7 sylXanc.6 . 2 (𝜑𝜁)
8 syl321anc.7 . 2 (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ 𝜁) → 𝜎)
91, 2, 3, 6, 7, 8syl311anc 1184 1 (𝜑𝜎)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-3an 922
This theorem is referenced by:  syl322anc  1198
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