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Metamath Proof Explorer

Theorem r19.21

Description: Restricted quantifier version of 19.21 . (Contributed by Scott Fenton, 30-Mar-2011)

Ref Expression
Hypothesis r19.21.1 ⊢ Ⅎ x φ
Assertion r19.21 ⊢ ∀ x ∈ A φ → ψ ↔ φ → ∀ x ∈ A ψ

Proof

Step Hyp Ref Expression
1 r19.21.1 ⊢ Ⅎ x φ
2 r19.21t ⊢ Ⅎ x φ → ∀ x ∈ A φ → ψ ↔ φ → ∀ x ∈ A ψ
3 1 2 ax-mp ⊢ ∀ x ∈ A φ → ψ ↔ φ → ∀ x ∈ A ψ