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	⊢ ( 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		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
3	1	imim2i	⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( 𝑥 ∈ 𝐴 → ( 𝜓 ↔ 𝜃 ) ) )
4	3	pm5.32d	⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜃 ) ) )
5	4	alexbii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜃 ) ) )
6	2 5	sylbi	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜃 ) ) )
7		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
8		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜃 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜃 ) )
9	6 7 8	3bitr4g	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜃 ) )