Theorem bj-wnfnf

Description: When ph is substituted for ps , this statement expresses nonfreeness in the weak form of nonfreeness ( E. -> A. ) . Note that this could also be proved from bj-nnfim , bj-nnfe1 and bj-nnfa1 . (Contributed by BJ, 9-Dec-2023)

	Ref	Expression
Assertion	bj-wnfnf	⊢ Ⅎ' 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-wnf2	⊢ ( ∃ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )
2		bj-wnf1	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )
3		df-bj-nnf	⊢ ( Ⅎ' 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ↔ ( ( ∃ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) ∧ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) ) )
4	1 2 3	mpbir2an	⊢ Ⅎ' 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 )