Theorem syl313anc
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
\( \vdash \varphi \rightarrow \sigma \)
syl313anc.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
\( \vdash \varphi \rightarrow \tau \)
l4
syl23anc.5
l5
syl33anc.6
syl133anc.7
l7
3jca l4 l5 _
\( \vdash \varphi \rightarrow \eta \wedge \zeta \wedge \sigma \)
syl313anc.8
\( \vdash (\psi \wedge \chi \wedge \theta) \wedge \tau \wedge (\eta \wedge \zeta \wedge \sigma) \rightarrow \rho \)
syl311anc l0 l1 l2 l3 l7 _