Theorem syl131anc
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
\( \vdash \varphi \rightarrow \eta \)
syl131anc.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
l2
syl3anc.3
syl3Xanc.4
l4
3jca l1 l2 _
\( \vdash \varphi \rightarrow \chi \wedge \theta \wedge \tau \)
l5
syl23anc.5
\( \vdash \varphi \rightarrow \eta \)
syl131anc.6
\( \vdash \psi \wedge (\chi \wedge \theta \wedge \tau) \wedge \eta \rightarrow \zeta \)
syl3anc l0 l4 l5 _