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	|- ( ( ( E. x ph -> A. x ph ) \/ ( E. x ps -> A. x ps ) ) -> ( ( E. x ph -> A. x ps ) <-> A. x ( ph -> ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		19.38	|- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
2		bj-alrimg	|- ( ( E. x ph -> A. x ph ) -> ( A. x ( ph -> ps ) -> ( E. x ph -> A. x ps ) ) )
3		bj-exlimg	|- ( ( E. x ps -> A. x ps ) -> ( A. x ( ph -> ps ) -> ( E. x ph -> A. x ps ) ) )
4	2 3	jaoi	|- ( ( ( E. x ph -> A. x ph ) \/ ( E. x ps -> A. x ps ) ) -> ( A. x ( ph -> ps ) -> ( E. x ph -> A. x ps ) ) )
5	1 4	impbid2	|- ( ( ( E. x ph -> A. x ph ) \/ ( E. x ps -> A. x ps ) ) -> ( ( E. x ph -> A. x ps ) <-> A. x ( ph -> ps ) ) )