Theorem bj-pm11.53vw

Description: Version of pm11.53v with nonfreeness antecedents. One can also prove the theorem with antecedent ( F// y A. x ph /\ A. y F// x ps ) . (Contributed by BJ, 7-Oct-2024)

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

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 ∧ Ⅎ' 𝑥 ∀ 𝑦 𝜓 ) → ∀ 𝑥 Ⅎ' 𝑦 𝜑 )
2		bj-19.21t	⊢ ( Ⅎ' 𝑦 𝜑 → ( ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑦 𝜓 ) ) )
3	1 2	sylg	⊢ ( ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 ∧ Ⅎ' 𝑥 ∀ 𝑦 𝜓 ) → ∀ 𝑥 ( ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑦 𝜓 ) ) )
4		albi	⊢ ( ∀ 𝑥 ( ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑦 𝜓 ) ) → ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜓 ) ) )
5	3 4	syl	⊢ ( ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 ∧ Ⅎ' 𝑥 ∀ 𝑦 𝜓 ) → ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜓 ) ) )
6		bj-19.23t	⊢ ( Ⅎ' 𝑥 ∀ 𝑦 𝜓 → ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )
7	6	adantl	⊢ ( ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 ∧ Ⅎ' 𝑥 ∀ 𝑦 𝜓 ) → ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )
8	5 7	bitrd	⊢ ( ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 ∧ Ⅎ' 𝑥 ∀ 𝑦 𝜓 ) → ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )