New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  syl223anc GIF version

Theorem syl223anc 1208
 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)
syl223anc.8 (((ψ χ) (θ τ) (η ζ σ)) → ρ)
Assertion
Ref Expression
syl223anc (φρ)

Proof of Theorem syl223anc
StepHypRef Expression
1 sylXanc.1 . 2 (φψ)
2 sylXanc.2 . 2 (φχ)
3 sylXanc.3 . . 3 (φθ)
4 sylXanc.4 . . 3 (φτ)
53, 4jca 518 . 2 (φ → (θ τ))
6 sylXanc.5 . 2 (φη)
7 sylXanc.6 . 2 (φζ)
8 sylXanc.7 . 2 (φσ)
9 syl223anc.8 . 2 (((ψ χ) (θ τ) (η ζ σ)) → ρ)
101, 2, 5, 6, 7, 8, 9syl213anc 1201 1 (φρ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator