Theorem bj-nfimexal

Description: A weak from of nonfreeness in either an antecedent or a consequent implies that a universally quantified implication is equivalent to the associated implication where the antecedent is existentially quantified and the consequent is universally quantified. The forward implication always holds (this is 19.38 ) and the converse implication is the join of instances of bj-alrimg and bj-exlimg (see 19.38a and 19.38b ). TODO: prove a version where the antecedents use the nonfreeness quantifier. (Contributed by BJ, 9-Dec-2023)

	Ref	Expression
Assertion	bj-nfimexal	$$⊢\left(\left(\exists x\phi \to \forall x\phi \right)\vee \left(\exists x\psi \to \forall x\psi \right)\right)\to \left(\left(\exists x\phi \to \forall x\psi \right)\leftrightarrow \forall x\left(\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		19.38	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \forall x\left(\phi \to \psi \right)$$
2		bj-alrimg	$$⊢\left(\exists x\phi \to \forall x\phi \right)\to \left(\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \forall x\psi \right)\right)$$
3		bj-exlimg	$$⊢\left(\exists x\psi \to \forall x\psi \right)\to \left(\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \forall x\psi \right)\right)$$
4	2 3	jaoi	$$⊢\left(\left(\exists x\phi \to \forall x\phi \right)\vee \left(\exists x\psi \to \forall x\psi \right)\right)\to \left(\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \forall x\psi \right)\right)$$
5	1 4	impbid2	$$⊢\left(\left(\exists x\phi \to \forall x\phi \right)\vee \left(\exists x\psi \to \forall x\psi \right)\right)\to \left(\left(\exists x\phi \to \forall x\psi \right)\leftrightarrow \forall x\left(\phi \to \psi \right)\right)$$