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Theorem syl233anc 1199
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)
sylXanc.8 (𝜑𝜌)
syl233anc.9 (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)
Assertion
Ref Expression
syl233anc (𝜑𝜇)

Proof of Theorem syl233anc
StepHypRef Expression
1 sylXanc.1 . . 3 (𝜑𝜓)
2 sylXanc.2 . . 3 (𝜑𝜒)
31, 2jca 300 . 2 (𝜑 → (𝜓𝜒))
4 sylXanc.3 . 2 (𝜑𝜃)
5 sylXanc.4 . 2 (𝜑𝜏)
6 sylXanc.5 . 2 (𝜑𝜂)
7 sylXanc.6 . 2 (𝜑𝜁)
8 sylXanc.7 . 2 (𝜑𝜎)
9 sylXanc.8 . 2 (𝜑𝜌)
10 syl233anc.9 . 2 (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)
113, 4, 5, 6, 7, 8, 9, 10syl133anc 1193 1 (𝜑𝜇)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  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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