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Theorem syl312anc 1156
 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)
syl312anc.7 (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁)) → 𝜎)
Assertion
Ref Expression
syl312anc (𝜑𝜎)

Proof of Theorem syl312anc
StepHypRef Expression
1 sylXanc.1 . 2 (𝜑𝜓)
2 sylXanc.2 . 2 (𝜑𝜒)
3 sylXanc.3 . 2 (𝜑𝜃)
4 sylXanc.4 . 2 (𝜑𝜏)
5 sylXanc.5 . . 3 (𝜑𝜂)
6 sylXanc.6 . . 3 (𝜑𝜁)
75, 6jca 290 . 2 (𝜑 → (𝜂𝜁))
8 syl312anc.7 . 2 (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁)) → 𝜎)
91, 2, 3, 4, 7, 8syl311anc 1149 1 (𝜑𝜎)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101 This theorem depends on definitions:  df-bi 110  df-3an 887 This theorem is referenced by: (None)
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