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Metamath Proof Explorer

Theorem rexbii

Description: Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994) (Revised by Mario Carneiro, 17-Oct-2016) (Proof shortened by Wolf Lammen, 6-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		rexbii.1	⊢ ( 𝜑 ↔ 𝜓 )
2	1	a1i	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3	2	rexbiia	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 𝜓 )