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Theorem syl31anc 1185
 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)
syl31anc.5 (((ψ χ θ) τ) → η)
Assertion
Ref Expression
syl31anc (φη)

Proof of Theorem syl31anc
StepHypRef Expression
1 sylXanc.1 . . 3 (φψ)
2 sylXanc.2 . . 3 (φχ)
3 sylXanc.3 . . 3 (φθ)
41, 2, 33jca 1132 . 2 (φ → (ψ χ θ))
5 sylXanc.4 . 2 (φτ)
6 syl31anc.5 . 2 (((ψ χ θ) τ) → η)
74, 5, 6syl2anc 642 1 (φη)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936 This theorem is referenced by:  syl32anc  1190  ltfinirr  4457  vfin1cltv  4547  leconnnc  6218  tlecg  6230
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