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Theorem syl232anc 1162
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)
syl232anc.8 (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎)) → 𝜌)
Assertion
Ref Expression
syl232anc (𝜑𝜌)

Proof of Theorem syl232anc
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 . . 3 (𝜑𝜁)
7 sylXanc.7 . . 3 (𝜑𝜎)
86, 7jca 290 . 2 (𝜑 → (𝜁𝜎))
9 syl232anc.8 . 2 (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎)) → 𝜌)
101, 2, 3, 4, 5, 8, 9syl231anc 1155 1 (𝜑𝜌)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  w3a 885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110  df-3an 887
This theorem is referenced by: (None)
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