Theorem r19.23t

Description: Closed theorem form of r19.23 . (Contributed by NM, 4-Mar-2013) (Revised by Mario Carneiro, 8-Oct-2016)

	Ref	Expression
Assertion	r19.23t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		19.23t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ) )
2		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
3		impexp	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
4	3	albii	⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
5	2 4	bitr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) )
6		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
7	6	imbi1i	⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) )
8	1 5 7	3bitr4g	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) ) )