Theorem issetf 2767

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.)

Hypothesis

Ref	Expression
issetf.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
issetf	⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Proof of Theorem issetf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isset 2766	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2		issetf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	nfeq2 2348	. . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴
4		nfv 1539	. . 3 ⊢ Ⅎ𝑦 𝑥 = 𝐴
5		eqeq1 2200	. . 3 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴))
6	3, 4, 5	cbvex 1767	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
7	1, 6	bitri 184	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  Ⅎwnfc 2323  Vcvv 2760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762

This theorem is referenced by:  vtoclgf  2818  spcimgft  2836  spcimegft  2838  bj-vtoclgft  15267