Theorem rexbiia

Description: Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999)

	Ref	Expression
Hypothesis	rexbiia.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	rexbiia	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		rexbiia.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2	1	pm5.32i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
3	2	rexbii2	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 𝜓 )