Theorem ralrexbid

Description: Formula-building rule for restricted existential quantifier, using a restricted universal quantifier to bind the quantified variable in the antecedent. (Contributed by AV, 21-Oct-2023) Reduce axiom usage. (Revised by SN, 13-Nov-2023)

	Ref	Expression
Hypothesis	ralrexbid.1	|- ( ph -> ( ps <-> th ) )
Assertion	ralrexbid	|- ( A. x e. A ph -> ( E. x e. A ps <-> E. x e. A th ) )

Proof

Step	Hyp	Ref	Expression
1		ralrexbid.1	|- ( ph -> ( ps <-> th ) )
2		df-ral	|- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
3	1	imim2i	|- ( ( x e. A -> ph ) -> ( x e. A -> ( ps <-> th ) ) )
4	3	pm5.32d	|- ( ( x e. A -> ph ) -> ( ( x e. A /\ ps ) <-> ( x e. A /\ th ) ) )
5	4	alexbii	|- ( A. x ( x e. A -> ph ) -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. A /\ th ) ) )
6	2 5	sylbi	|- ( A. x e. A ph -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. A /\ th ) ) )
7		df-rex	|- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
8		df-rex	|- ( E. x e. A th <-> E. x ( x e. A /\ th ) )
9	6 7 8	3bitr4g	|- ( A. x e. A ph -> ( E. x e. A ps <-> E. x e. A th ) )