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Metamath Proof Explorer

Theorem bj-19.37im

Description: One direction of 19.37 from the same axioms as 19.37imv . (Contributed by BJ, 2-Dec-2023)

Ref Expression
Assertion bj-19.37im ⊢ Ⅎ' x φ → ∃ x φ → ψ → φ → ∃ x ψ

Proof

Step Hyp Ref Expression
1 19.35 ⊢ ∃ x φ → ψ ↔ ∀ x φ → ∃ x ψ
2 bj-nnfa ⊢ Ⅎ' x φ → φ → ∀ x φ
3 2 imim1d ⊢ Ⅎ' x φ → ∀ x φ → ∃ x ψ → φ → ∃ x ψ
4 1 3 biimtrid ⊢ Ⅎ' x φ → ∃ x φ → ψ → φ → ∃ x ψ