Theorem rexsns

Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015) (Revised by NM, 22-Aug-2018)

	Ref	Expression
Assertion	rexsns	⊢ ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		velsn	⊢ ( 𝑥 ∈ { 𝐴 } ↔ 𝑥 = 𝐴 )
2	1	anbi1i	⊢ ( ( 𝑥 ∈ { 𝐴 } ∧ 𝜑 ) ↔ ( 𝑥 = 𝐴 ∧ 𝜑 ) )
3	2	exbii	⊢ ( ∃ 𝑥 ( 𝑥 ∈ { 𝐴 } ∧ 𝜑 ) ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) )
4		df-rex	⊢ ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ { 𝐴 } ∧ 𝜑 ) )
5		sbc5	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) )
6	3 4 5	3bitr4i	⊢ ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 )