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Metamath Proof Explorer

Theorem r19.36vf

Description: Restricted quantifier version of one direction of 19.36 . (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypothesis r19.36vf.1 ⊢ Ⅎ 𝑥 𝜓
Assertion r19.36vf ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )

Proof

Step Hyp Ref Expression
1 r19.36vf.1 ⊢ Ⅎ 𝑥 𝜓
2 r19.35 ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
3 idd ⊢ ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜓 ) )
4 1 3 rexlimi ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜓 )
5 4 imim2i ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )
6 2 5 sylbi ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) )