Theorem aleximi

A variant of al2imi: instead of applying A. x quantifiers to the final implication, replace them with E. x. A shorter proof is possible using nfa1, sps and eximd, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)

Hypothesis
aleximi.1
\( \vdash \varphi \rightarrow \psi \rightarrow \chi \)
Conclusion
\( \vdash \forall x \, \varphi \rightarrow \exists x \, \psi \rightarrow \exists x \, \chi \)
Proof
aleximi.1
con3d _
l2
al2imi _
\( \vdash \forall x \, \varphi \rightarrow \forall x \, \neg \chi \rightarrow \forall x \, \neg \psi \)
l3
alnex
\( \vdash \forall x \, \neg \chi \leftrightarrow \neg \exists x \, \chi \)
alnex
\( \vdash \forall x \, \neg \psi \leftrightarrow \neg \exists x \, \psi \)
3imtr3g l2 l3 _
con4d _