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Metamath Proof Explorer

Theorem reueqdv

Description: Formula-building rule for restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 1-Sep-2025)

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

Proof

Step Hyp Ref Expression
1 reueqdv.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 reueq1 ⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐵 𝜓 ) )
3 1 2 syl ⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐵 𝜓 ) )