Theorem syl33anc
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
\( \vdash \varphi \rightarrow \zeta \)
syl33anc.7
\( \vdash (\psi \wedge \chi \wedge \theta) \wedge (\tau \wedge \eta \wedge \zeta) \rightarrow \sigma \)
Conclusion
\( \vdash \varphi \rightarrow \sigma \)
Proof
l0
syl3anc.1
l1
syl3anc.2
syl3anc.3
l3
3jca l0 l1 _
\( \vdash \varphi \rightarrow \psi \wedge \chi \wedge \theta \)
l4
syl3Xanc.4
\( \vdash \varphi \rightarrow \tau \)
l5
syl23anc.5
\( \vdash \varphi \rightarrow \eta \)
l6
syl33anc.6
\( \vdash \varphi \rightarrow \zeta \)
syl33anc.7
\( \vdash (\psi \wedge \chi \wedge \theta) \wedge (\tau \wedge \eta \wedge \zeta) \rightarrow \sigma \)
syl13anc l3 l4 l5 l6 _