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Metamath Proof Explorer

Theorem reubii

Description: Formula-building rule for restricted existential uniqueness quantifier (inference form). (Contributed by NM, 22-Oct-1999)

Ref Expression
Hypothesis rmobii.1 ⊢ ( 𝜑 ↔ 𝜓 )
Assertion reubii ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐴 𝜓 )

Proof

Step Hyp Ref Expression
1 rmobii.1 ⊢ ( 𝜑 ↔ 𝜓 )
2 1 a1i ⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3 2 reubiia ⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ∈ 𝐴 𝜓 )