Theorem 19.40b

Description: The antecedent provides a condition implying the converse of 19.40 . This is to 19.40 what 19.33b is to 19.33 . (Contributed by BJ, 6-May-2019) (Proof shortened by Wolf Lammen, 13-Nov-2020)

	Ref	Expression
Assertion	19.40b	|- ( ( A. x ph \/ A. x ps ) -> ( ( E. x ph /\ E. x ps ) <-> E. x ( ph /\ ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm3.21	|- ( ps -> ( ph -> ( ph /\ ps ) ) )
2	1	aleximi	|- ( A. x ps -> ( E. x ph -> E. x ( ph /\ ps ) ) )
3		pm3.2	|- ( ph -> ( ps -> ( ph /\ ps ) ) )
4	3	aleximi	|- ( A. x ph -> ( E. x ps -> E. x ( ph /\ ps ) ) )
5	2 4	jaoa	|- ( ( A. x ps \/ A. x ph ) -> ( ( E. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) ) )
6	5	orcoms	|- ( ( A. x ph \/ A. x ps ) -> ( ( E. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) ) )
7		19.40	|- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) )
8	6 7	impbid1	|- ( ( A. x ph \/ A. x ps ) -> ( ( E. x ph /\ E. x ps ) <-> E. x ( ph /\ ps ) ) )