Theorem aleximi

Description: 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)

	Ref	Expression
Hypothesis	aleximi.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
Assertion	aleximi	$$⊢\forall x\phi \to \left(\exists x\psi \to \exists x\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		aleximi.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2	1	con3d	$$⊢\phi \to \left(\neg \chi \to \neg \psi \right)$$
3	2	al2imi	$$⊢\forall x\phi \to \left(\forall x\neg \chi \to \forall x\neg \psi \right)$$
4		alnex	$$⊢\forall x\neg \chi \leftrightarrow \neg \exists x\chi$$
5		alnex	$$⊢\forall x\neg \psi \leftrightarrow \neg \exists x\psi$$
6	3 4 5	3imtr3g	$$⊢\forall x\phi \to \left(\neg \exists x\chi \to \neg \exists x\psi \right)$$
7	6	con4d	$$⊢\forall x\phi \to \left(\exists x\psi \to \exists x\chi \right)$$