Theorem bj-19.42t

Description: Closed form of 19.42 from the same axioms as 19.42v . (Contributed by BJ, 2-Dec-2023)

	Ref	Expression
Assertion	bj-19.42t	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		19.40	⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) → ( ∃ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) )
2		bj-nnfe	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → 𝜑 ) )
3	2	anim1d	⊢ ( Ⅎ' 𝑥 𝜑 → ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) → ( 𝜑 ∧ ∃ 𝑥 𝜓 ) ) )
4	1 3	syl5	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ ∃ 𝑥 𝜓 ) ) )
5		bj-nnfa	⊢ ( Ⅎ' 𝑥 𝜑 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
6	5	anim1d	⊢ ( Ⅎ' 𝑥 𝜑 → ( ( 𝜑 ∧ ∃ 𝑥 𝜓 ) → ( ∀ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) ) )
7		19.29	⊢ ( ( ∀ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )
8	6 7	syl6	⊢ ( Ⅎ' 𝑥 𝜑 → ( ( 𝜑 ∧ ∃ 𝑥 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
9	4 8	impbid	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 𝜓 ) ) )