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Metamath Proof Explorer

Theorem bj-wnf2

Description: When ph is substituted for ps , this is the first half of nonfreness ( . -> A. ) of the weak form of nonfreeness ( E. -> A. ) . (Contributed by BJ, 9-Dec-2023)

Ref Expression
Assertion bj-wnf2 ⊢ ∃ x ∃ x φ → ∀ x ψ → ∃ x φ → ∀ x ψ

Proof

Step Hyp Ref Expression
1 hbe1 ⊢ ∃ x φ → ∀ x ∃ x φ
2 bj-eximcom ⊢ ∃ x ∃ x φ → ∀ x ψ → ∀ x ∃ x φ → ∃ x ∀ x ψ
3 hbe1a ⊢ ∃ x ∀ x ψ → ∀ x ψ
4 1 2 3 syl56 ⊢ ∃ x ∃ x φ → ∀ x ψ → ∃ x φ → ∀ x ψ