Theorem syl213anc
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
\( \vdash \varphi \rightarrow \zeta \)
syl213anc.7
\( \vdash \psi \wedge \chi \wedge \theta \wedge (\tau \wedge \eta \wedge \zeta) \rightarrow \sigma \)
Conclusion
\( \vdash \varphi \rightarrow \sigma \)
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
\( \vdash \varphi \rightarrow \tau \)
l5
syl23anc.5
\( \vdash \varphi \rightarrow \eta \)
l6
syl33anc.6
\( \vdash \varphi \rightarrow \zeta \)
syl213anc.7
\( \vdash \psi \wedge \chi \wedge \theta \wedge (\tau \wedge \eta \wedge \zeta) \rightarrow \sigma \)
syl113anc l2 l3 l4 l5 l6 _