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Theorem syl131anc 1233
 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)
syl131anc.6 ((𝜓 ∧ (𝜒𝜃𝜏) ∧ 𝜂) → 𝜁)
Assertion
Ref Expression
syl131anc (𝜑𝜁)

Proof of Theorem syl131anc
StepHypRef Expression
1 sylXanc.1 . 2 (𝜑𝜓)
2 sylXanc.2 . . 3 (𝜑𝜒)
3 sylXanc.3 . . 3 (𝜑𝜃)
4 sylXanc.4 . . 3 (𝜑𝜏)
52, 3, 43jca 1162 . 2 (𝜑 → (𝜒𝜃𝜏))
6 sylXanc.5 . 2 (𝜑𝜂)
7 syl131anc.6 . 2 ((𝜓 ∧ (𝜒𝜃𝜏) ∧ 𝜂) → 𝜁)
81, 5, 6, 7syl3anc 1220 1 (𝜑𝜁)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ w3a 963 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 965 This theorem is referenced by:  syl132anc  1238  syl231anc  1240  syl133anc  1243
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