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Theorem syl231anc 1221
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)
syl231anc.7 (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ 𝜁) → 𝜎)
Assertion
Ref Expression
syl231anc (𝜑𝜎)

Proof of Theorem syl231anc
StepHypRef Expression
1 sylXanc.1 . . 3 (𝜑𝜓)
2 sylXanc.2 . . 3 (𝜑𝜒)
31, 2jca 304 . 2 (𝜑 → (𝜓𝜒))
4 sylXanc.3 . 2 (𝜑𝜃)
5 sylXanc.4 . 2 (𝜑𝜏)
6 sylXanc.5 . 2 (𝜑𝜂)
7 sylXanc.6 . 2 (𝜑𝜁)
8 syl231anc.7 . 2 (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ 𝜁) → 𝜎)
93, 4, 5, 6, 7, 8syl131anc 1214 1 (𝜑𝜎)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  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:  syl232anc  1228  prarloc  7279
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