Theorem dfrals2

Description: The bounded "all some" form is the general form with the class membership folded into the antecedent. (Contributed by David A. Wheeler, 22-Oct-2018) (Revised by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Assertion	dfrals2	|- ( AE x e. A ( ph -> ps ) <-> AE x ( ( x e. A /\ ph ) -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		df-ral	|- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
2		impexp	|- ( ( ( x e. A /\ ph ) -> ps ) <-> ( x e. A -> ( ph -> ps ) ) )
3	2	albii	|- ( A. x ( ( x e. A /\ ph ) -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
4	1 3	bitr4i	|- ( A. x e. A ( ph -> ps ) <-> A. x ( ( x e. A /\ ph ) -> ps ) )
5		df-rex	|- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6	4 5	anbi12i	|- ( ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) <-> ( A. x ( ( x e. A /\ ph ) -> ps ) /\ E. x ( x e. A /\ ph ) ) )
7		df-rals	|- ( AE x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) )
8		df-als	|- ( AE x ( ( x e. A /\ ph ) -> ps ) <-> ( A. x ( ( x e. A /\ ph ) -> ps ) /\ E. x ( x e. A /\ ph ) ) )
9	6 7 8	3bitr4i	|- ( AE x e. A ( ph -> ps ) <-> AE x ( ( x e. A /\ ph ) -> ps ) )