Theorem bj-pm11.53v

Description: Version of pm11.53v with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024)

	Ref	Expression
Assertion	bj-pm11.53v	⊢ ( ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 ∧ ∀ 𝑦 Ⅎ' 𝑥 𝜓 ) → ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-nnfalt	⊢ ( ∀ 𝑦 Ⅎ' 𝑥 𝜓 → Ⅎ' 𝑥 ∀ 𝑦 𝜓 )
2		bj-pm11.53vw	⊢ ( ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 ∧ Ⅎ' 𝑥 ∀ 𝑦 𝜓 ) → ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )
3	1 2	sylan2	⊢ ( ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 ∧ ∀ 𝑦 Ⅎ' 𝑥 𝜓 ) → ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )