# Metamath Proof Explorer

## Theorem syl332anc

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012)

Ref Expression
Hypotheses syl3anc.1 ${⊢}{\phi }\to {\psi }$
syl3anc.2 ${⊢}{\phi }\to {\chi }$
syl3anc.3 ${⊢}{\phi }\to {\theta }$
syl3Xanc.4 ${⊢}{\phi }\to {\tau }$
syl23anc.5 ${⊢}{\phi }\to {\eta }$
syl33anc.6 ${⊢}{\phi }\to {\zeta }$
syl133anc.7 ${⊢}{\phi }\to {\sigma }$
syl233anc.8 ${⊢}{\phi }\to {\rho }$
syl332anc.9 ${⊢}\left(\left({\psi }\wedge {\chi }\wedge {\theta }\right)\wedge \left({\tau }\wedge {\eta }\wedge {\zeta }\right)\wedge \left({\sigma }\wedge {\rho }\right)\right)\to {\mu }$
Assertion syl332anc ${⊢}{\phi }\to {\mu }$

### Proof

Step Hyp Ref Expression
1 syl3anc.1 ${⊢}{\phi }\to {\psi }$
2 syl3anc.2 ${⊢}{\phi }\to {\chi }$
3 syl3anc.3 ${⊢}{\phi }\to {\theta }$
4 syl3Xanc.4 ${⊢}{\phi }\to {\tau }$
5 syl23anc.5 ${⊢}{\phi }\to {\eta }$
6 syl33anc.6 ${⊢}{\phi }\to {\zeta }$
7 syl133anc.7 ${⊢}{\phi }\to {\sigma }$
8 syl233anc.8 ${⊢}{\phi }\to {\rho }$
9 syl332anc.9 ${⊢}\left(\left({\psi }\wedge {\chi }\wedge {\theta }\right)\wedge \left({\tau }\wedge {\eta }\wedge {\zeta }\right)\wedge \left({\sigma }\wedge {\rho }\right)\right)\to {\mu }$
10 7 8 jca ${⊢}{\phi }\to \left({\sigma }\wedge {\rho }\right)$
11 1 2 3 4 5 6 10 9 syl331anc ${⊢}{\phi }\to {\mu }$