Theorem syl211anc

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
\( \vdash \varphi \rightarrow \tau \)
syl211anc.5
\( \vdash \psi \wedge \chi \wedge \theta \wedge \tau \rightarrow \eta \)
Conclusion
\( \vdash \varphi \rightarrow \eta \)
Proof
l0
syl3anc.1
syl3anc.2
l2
jca l0 _
\( \vdash \varphi \rightarrow \psi \wedge \chi \)
l3
syl3anc.3
\( \vdash \varphi \rightarrow \theta \)
l4
syl3Xanc.4
\( \vdash \varphi \rightarrow \tau \)
syl211anc.5
\( \vdash \psi \wedge \chi \wedge \theta \wedge \tau \rightarrow \eta \)
syl3anc l2 l3 l4 _