Theorem syl123anc
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
\( \vdash \varphi \rightarrow \zeta \)
syl123anc.7
\( \vdash \psi \wedge \chi \wedge \theta \wedge (\tau \wedge \eta \wedge \zeta) \rightarrow \sigma \)
Conclusion
\( \vdash \varphi \rightarrow \sigma \)
Proof
l0
syl3anc.1
\( \vdash \varphi \rightarrow \psi \)
l1
syl3anc.2
syl3anc.3
l3
jca l1 _
\( \vdash \varphi \rightarrow \chi \wedge \theta \)
l4
syl3Xanc.4
\( \vdash \varphi \rightarrow \tau \)
l5
syl23anc.5
\( \vdash \varphi \rightarrow \eta \)
l6
syl33anc.6
\( \vdash \varphi \rightarrow \zeta \)
syl123anc.7
\( \vdash \psi \wedge \chi \wedge \theta \wedge (\tau \wedge \eta \wedge \zeta) \rightarrow \sigma \)
syl113anc l0 l3 l4 l5 l6 _