Theorem syl331anc
Syllogism combined with contraction. (Contributed by NM,
11-Mar-2012.)
Hypotheses
syl3anc.1
\( \vdash \varphi \rightarrow \psi \)
syl3anc.2
\( \vdash \varphi \rightarrow \chi \)
syl3anc.3
\( \vdash \varphi \rightarrow \theta \)
syl3Xanc.4
\( \vdash \varphi \rightarrow \tau \)
syl23anc.5
\( \vdash \varphi \rightarrow \eta \)
syl33anc.6
\( \vdash \varphi \rightarrow \zeta \)
syl133anc.7
\( \vdash \varphi \rightarrow \sigma \)
syl331anc.8
\( \vdash (\psi \wedge \chi \wedge \theta) \wedge (\tau \wedge \eta \wedge \zeta) \wedge \sigma \rightarrow \rho \)
Conclusion
\( \vdash \varphi \rightarrow \rho \)
Proof
l0
syl3anc.1
\( \vdash \varphi \rightarrow \psi \)
l1
syl3anc.2
\( \vdash \varphi \rightarrow \chi \)
l2
syl3anc.3
\( \vdash \varphi \rightarrow \theta \)
l3
syl3Xanc.4
l4
syl23anc.5
syl33anc.6
l6
3jca l3 l4 _
\( \vdash \varphi \rightarrow \tau \wedge \eta \wedge \zeta \)
l7
syl133anc.7
\( \vdash \varphi \rightarrow \sigma \)
syl331anc.8
\( \vdash (\psi \wedge \chi \wedge \theta) \wedge (\tau \wedge \eta \wedge \zeta) \wedge \sigma \rightarrow \rho \)
syl311anc l0 l1 l2 l6 l7 _