Theorem bj-pm11.53a

Description: A variant of pm11.53v . One can similarly prove a variant with DV ( y , ph ) and A. y F// x ps instead of DV ( x , ps ) and A. x F// y ph . (Contributed by BJ, 7-Oct-2024)

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

Proof

Step	Hyp	Ref	Expression
1		bj-nnfv	⊢ 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	mpan2	⊢ ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 → ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )