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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 ) )