Theorem reueqd

Description: Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004)

	Ref	Expression
Hypothesis	rmoeqd.1	⊢ ( 𝐴 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
Assertion	reueqd	⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐵 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		rmoeqd.1	⊢ ( 𝐴 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
2		reueq1	⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐵 𝜑 ) )
3	1	reubidv	⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ∈ 𝐵 𝜑 ↔ ∃! 𝑥 ∈ 𝐵 𝜓 ) )
4	2 3	bitrd	⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐵 𝜓 ) )