Theorem syl212anc
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
\( \vdash \varphi \rightarrow \eta \)
syl212anc.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
\( \vdash \varphi \rightarrow \theta \)
l3
syl3Xanc.4
syl23anc.5
l5
jca l3 _
\( \vdash \varphi \rightarrow \tau \wedge \eta \)
syl212anc.6
\( \vdash \psi \wedge \chi \wedge \theta \wedge \tau \wedge \eta \rightarrow \zeta \)
syl211anc l0 l1 l2 l5 _