Theorem rexeqbi1dv

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997) (Proof shortened by Steven Nguyen, 5-May-2023)

	Ref	Expression
Hypothesis	raleqbi1dv.1	⊢ ( 𝐴 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
Assertion	rexeqbi1dv	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		raleqbi1dv.1	⊢ ( 𝐴 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
2		id	⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
3	2 1	rexeqbidvv	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜓 ) )