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) (Proof shortened by Wolf Lammen, 4-Nov-2024)

	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	1	ralimi	|- ( A. x e. A ph -> A. x e. A ( ps <-> th ) )
3		rexbi	|- ( A. x e. A ( ps <-> th ) -> ( E. x e. A ps <-> E. x e. A th ) )
4	2 3	syl	|- ( A. x e. A ph -> ( E. x e. A ps <-> E. x e. A th ) )