Theorem issetf

Description: A version of isset that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011) (Revised by Mario Carneiro, 10-Oct-2016)

	Ref	Expression
Hypothesis	issetf.1	⊢ Ⅎ 𝑥 𝐴
Assertion	issetf	⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		issetf.1	⊢ Ⅎ 𝑥 𝐴
2		isset	⊢ ( 𝐴 ∈ V ↔ ∃ 𝑦 𝑦 = 𝐴 )
3	1	nfeq2	⊢ Ⅎ 𝑥 𝑦 = 𝐴
4		nfv	⊢ Ⅎ 𝑦 𝑥 = 𝐴
5		eqeq1	⊢ ( 𝑦 = 𝑥 → ( 𝑦 = 𝐴 ↔ 𝑥 = 𝐴 ) )
6	3 4 5	cbvexv1	⊢ ( ∃ 𝑦 𝑦 = 𝐴 ↔ ∃ 𝑥 𝑥 = 𝐴 )
7	2 6	bitri	⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )