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Theorem syl331anc 1226
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)
syl331anc.8 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ 𝜎) → 𝜌)
Assertion
Ref Expression
syl331anc (𝜑𝜌)

Proof of Theorem syl331anc
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)
6 sylXanc.6 . . 3 (𝜑𝜁)
74, 5, 63jca 1146 . 2 (𝜑 → (𝜏𝜂𝜁))
8 sylXanc.7 . 2 (𝜑𝜎)
9 syl331anc.8 . 2 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ 𝜎) → 𝜌)
101, 2, 3, 7, 8, 9syl311anc 1215 1 (𝜑𝜌)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 947
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 949
This theorem is referenced by:  syl332anc  1232  syl333anc  1233  qredeu  11705
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