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Theorem syl113anc 1182
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)
syl113anc.6 ((𝜓𝜒 ∧ (𝜃𝜏𝜂)) → 𝜁)
Assertion
Ref Expression
syl113anc (𝜑𝜁)

Proof of Theorem syl113anc
StepHypRef Expression
1 sylXanc.1 . 2 (𝜑𝜓)
2 sylXanc.2 . 2 (𝜑𝜒)
3 sylXanc.3 . . 3 (𝜑𝜃)
4 sylXanc.4 . . 3 (𝜑𝜏)
5 sylXanc.5 . . 3 (𝜑𝜂)
63, 4, 53jca 1119 . 2 (𝜑 → (𝜃𝜏𝜂))
7 syl113anc.6 . 2 ((𝜓𝜒 ∧ (𝜃𝜏𝜂)) → 𝜁)
81, 2, 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:  syl123anc  1187  syl213anc  1189  divalglemnn  10525
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