Theorem reubiia

Description: Formula-building rule for restricted existential uniqueness quantifier (inference form). (Contributed by NM, 14-Nov-2004)

	Ref	Expression
Hypothesis	rmobiia.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	reubiia	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		rmobiia.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2	1	pm5.32i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
3	2	eubii	⊢ ( ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
4		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
5		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
6	3 4 5	3bitr4i	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐴 𝜓 )