Theorem syl322anc
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
\( \vdash \varphi \rightarrow \sigma \)
syl322anc.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
\( \vdash \varphi \rightarrow \eta \)
l5
syl33anc.6
syl133anc.7
l7
jca l5 _
\( \vdash \varphi \rightarrow \zeta \wedge \sigma \)
syl322anc.8
\( \vdash (\psi \wedge \chi \wedge \theta) \wedge \tau \wedge \eta \wedge \zeta \wedge \sigma \rightarrow \rho \)
syl321anc l0 l1 l2 l3 l4 l7 _