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Metamath Proof Explorer

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 ) )