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Theorem syl333anc 1202
 Description: A syllogism inference combined with contraction. (Contributed by NM, 10-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)
sylXanc.8 (𝜑𝜌)
sylXanc.9 (𝜑𝜇)
syl333anc.10 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌𝜇)) → 𝜆)
Assertion
Ref Expression
syl333anc (𝜑𝜆)

Proof of Theorem syl333anc
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 . 2 (𝜑𝜂)
6 sylXanc.6 . 2 (𝜑𝜁)
7 sylXanc.7 . . 3 (𝜑𝜎)
8 sylXanc.8 . . 3 (𝜑𝜌)
9 sylXanc.9 . . 3 (𝜑𝜇)
107, 8, 93jca 1119 . 2 (𝜑 → (𝜎𝜌𝜇))
11 syl333anc.10 . 2 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌𝜇)) → 𝜆)
121, 2, 3, 4, 5, 6, 10, 11syl331anc 1195 1 (𝜑𝜆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ 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: (None)
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