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Metamath Proof Explorer

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