Theorem bj-alexim

Description: Closed form of aleximi . Note: this proof is shorter, so aleximi could be deduced from it ( exim would have to be proved first, see bj-exim ). (Contributed by BJ, 8-Nov-2021)

	Ref	Expression
Assertion	bj-alexim	$$⊢\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\forall x\phi \to \left(\exists x\psi \to \exists x\chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		alim	$$⊢\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\forall x\phi \to \forall x\left(\psi \to \chi \right)\right)$$
2		exim	$$⊢\forall x\left(\psi \to \chi \right)\to \left(\exists x\psi \to \exists x\chi \right)$$
3	1 2	syl6	$$⊢\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\forall x\phi \to \left(\exists x\psi \to \exists x\chi \right)\right)$$