Theorem rexeqdv

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007)

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

Proof

Step	Hyp	Ref	Expression
1		raleqdv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		rexeq	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐵 𝜓 ) )
3	1 2	syl	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐵 𝜓 ) )