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Metamath Proof Explorer

Theorem 19.37

Description: Theorem 19.37 of Margaris p. 90. See 19.37v for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993)

Ref Expression
Hypothesis 19.37.1 ⊢ Ⅎ x φ
Assertion 19.37 ⊢ ∃ x φ → ψ ↔ φ → ∃ x ψ

Proof

Step Hyp Ref Expression
1 19.37.1 ⊢ Ⅎ x φ
2 19.35 ⊢ ∃ x φ → ψ ↔ ∀ x φ → ∃ x ψ
3 1 19.3 ⊢ ∀ x φ ↔ φ
4 3 imbi1i ⊢ ∀ x φ → ∃ x ψ ↔ φ → ∃ x ψ
5 2 4 bitri ⊢ ∃ x φ → ψ ↔ φ → ∃ x ψ