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Metamath Proof Explorer

Theorem ralsn

Description: Convert a universal quantification restricted to a singleton to a substitution. (Contributed by NM, 27-Apr-2009)

Ref Expression
Hypotheses ralsn.1 ⊢ A ∈ V
ralsn.2 ⊢ x = A → φ ↔ ψ
Assertion ralsn ⊢ ∀ x ∈ A φ ↔ ψ

Proof

Step Hyp Ref Expression
1 ralsn.1 ⊢ A ∈ V
2 ralsn.2 ⊢ x = A → φ ↔ ψ
3 2 ralsng ⊢ A ∈ V → ∀ x ∈ A φ ↔ ψ
4 1 3 ax-mp ⊢ ∀ x ∈ A φ ↔ ψ