Theorem reubidv

Description: Formula-building rule for restricted existential uniqueness quantifier (deduction form). (Contributed by NM, 17-Oct-1996)

	Ref	Expression
Hypothesis	rmobidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	reubidv	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐴 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rmobidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	reubidva	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐴 𝜒 ) )