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	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
Assertion	ralrexbid	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		ralrexbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
2	1	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜓 ↔ 𝜃 ) )
3		rexbi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜓 ↔ 𝜃 ) → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜃 ) )
4	2 3	syl	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜃 ) )