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Metamath Proof Explorer

Theorem r19.37v

Description: Restricted quantifier version of one direction of 19.37v . (The other direction holds iff A is nonempty, see r19.37zv .) (Contributed by NM, 2-Apr-2004) Reduce axiom usage. (Revised by Wolf Lammen, 18-Jun-2023)

Ref Expression
Assertion r19.37v ⊢ ∃ x ∈ A φ → ψ → φ → ∃ x ∈ A ψ

Proof

Step Hyp Ref Expression
1 id ⊢ φ → φ
2 1 ralrimivw ⊢ φ → ∀ x ∈ A φ
3 r19.35 ⊢ ∃ x ∈ A φ → ψ ↔ ∀ x ∈ A φ → ∃ x ∈ A ψ
4 3 biimpi ⊢ ∃ x ∈ A φ → ψ → ∀ x ∈ A φ → ∃ x ∈ A ψ
5 2 4 syl5 ⊢ ∃ x ∈ A φ → ψ → φ → ∃ x ∈ A ψ