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Theorem syl311anc 1184
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)
syl311anc.6 (((𝜓𝜒𝜃) ∧ 𝜏𝜂) → 𝜁)
Assertion
Ref Expression
syl311anc (𝜑𝜁)

Proof of Theorem syl311anc
StepHypRef Expression
1 sylXanc.1 . . 3 (𝜑𝜓)
2 sylXanc.2 . . 3 (𝜑𝜒)
3 sylXanc.3 . . 3 (𝜑𝜃)
41, 2, 33jca 1119 . 2 (𝜑 → (𝜓𝜒𝜃))
5 sylXanc.4 . 2 (𝜑𝜏)
6 sylXanc.5 . 2 (𝜑𝜂)
7 syl311anc.6 . 2 (((𝜓𝜒𝜃) ∧ 𝜏𝜂) → 𝜁)
84, 5, 6, 7syl3anc 1170 1 (𝜑𝜁)
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:  syl312anc  1191  syl321anc  1192  syl313anc  1194  syl331anc  1195
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