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Theorem syl313anc 1241
 Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
Hypotheses
Ref Expression
sylXanc.1 (𝜑𝜓)
sylXanc.2 (𝜑𝜒)
sylXanc.3 (𝜑𝜃)
sylXanc.4 (𝜑𝜏)
sylXanc.5 (𝜑𝜂)
sylXanc.6 (𝜑𝜁)
sylXanc.7 (𝜑𝜎)
syl313anc.8 (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁𝜎)) → 𝜌)
Assertion
Ref Expression
syl313anc (𝜑𝜌)

Proof of Theorem syl313anc
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)
7 sylXanc.7 . . 3 (𝜑𝜎)
85, 6, 73jca 1162 . 2 (𝜑 → (𝜂𝜁𝜎))
9 syl313anc.8 . 2 (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁𝜎)) → 𝜌)
101, 2, 3, 4, 8, 9syl311anc 1231 1 (𝜑𝜌)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ w3a 963 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107 This theorem depends on definitions:  df-bi 116  df-3an 965 This theorem is referenced by:  syl323anc  1247
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