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Theorem syl123anc 1216
 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)
syl123anc.7 ((𝜓 ∧ (𝜒𝜃) ∧ (𝜏𝜂𝜁)) → 𝜎)
Assertion
Ref Expression
syl123anc (𝜑𝜎)

Proof of Theorem syl123anc
StepHypRef Expression
1 sylXanc.1 . 2 (𝜑𝜓)
2 sylXanc.2 . . 3 (𝜑𝜒)
3 sylXanc.3 . . 3 (𝜑𝜃)
42, 3jca 302 . 2 (𝜑 → (𝜒𝜃))
5 sylXanc.4 . 2 (𝜑𝜏)
6 sylXanc.5 . 2 (𝜑𝜂)
7 sylXanc.6 . 2 (𝜑𝜁)
8 syl123anc.7 . 2 ((𝜓 ∧ (𝜒𝜃) ∧ (𝜏𝜂𝜁)) → 𝜎)
91, 4, 5, 6, 7, 8syl113anc 1211 1 (𝜑𝜎)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 945 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107 This theorem depends on definitions:  df-bi 116  df-3an 947 This theorem is referenced by: (None)
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