Theorem syl32anc
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
\( \vdash \varphi \rightarrow \eta \)
syl32anc.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 _
\( \vdash \varphi \rightarrow \tau \wedge \eta \)
syl32anc.6
\( \vdash (\psi \wedge \chi \wedge \theta) \wedge (\tau \wedge \eta) \rightarrow \zeta \)
syl31anc l0 l1 l2 l5 _