Theorem r19.12sn

Description: Special case of r19.12 where its converse holds. (Contributed by NM, 19-May-2008) (Revised by Mario Carneiro, 23-Apr-2015) (Revised by BJ, 18-Mar-2020)

	Ref	Expression
Assertion	r19.12sn	⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ∈ { 𝐴 } ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ { 𝐴 } 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sbcralg	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 [ 𝐴 / 𝑥 ] 𝜑 ) )
2		rexsns	⊢ ( ∃ 𝑥 ∈ { 𝐴 } ∀ 𝑦 ∈ 𝐵 𝜑 ↔ [ 𝐴 / 𝑥 ] ∀ 𝑦 ∈ 𝐵 𝜑 )
3		rexsns	⊢ ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 )
4	3	ralbii	⊢ ( ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 [ 𝐴 / 𝑥 ] 𝜑 )
5	1 2 4	3bitr4g	⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ∈ { 𝐴 } ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ { 𝐴 } 𝜑 ) )