Theorem syl113anc
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
\( \vdash \varphi \rightarrow \eta \)
syl113anc.6
\( \vdash \psi \wedge \chi \wedge (\theta \wedge \tau \wedge \eta) \rightarrow \zeta \)
Conclusion
\( \vdash \varphi \rightarrow \zeta \)
Proof
l0
syl3anc.1
\( \vdash \varphi \rightarrow \psi \)
l1
syl3anc.2
\( \vdash \varphi \rightarrow \chi \)
l2
syl3anc.3
l3
syl3Xanc.4
syl23anc.5
l5
3jca l2 l3 _
\( \vdash \varphi \rightarrow \theta \wedge \tau \wedge \eta \)
syl113anc.6
\( \vdash \psi \wedge \chi \wedge (\theta \wedge \tau \wedge \eta) \rightarrow \zeta \)
syl3anc l0 l1 l5 _