Theorem syl312anc
Syllogism combined with contraction. (Contributed by NM,
11-Jul-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
\( \vdash \varphi \rightarrow \zeta \)
syl312anc.7
\( \vdash (\psi \wedge \chi \wedge \theta) \wedge \tau \wedge \eta \wedge \zeta \rightarrow \sigma \)
Conclusion
\( \vdash \varphi \rightarrow \sigma \)
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
syl33anc.6
l6
jca l4 _
\( \vdash \varphi \rightarrow \eta \wedge \zeta \)
syl312anc.7
\( \vdash (\psi \wedge \chi \wedge \theta) \wedge \tau \wedge \eta \wedge \zeta \rightarrow \sigma \)
syl311anc l0 l1 l2 l3 l6 _