Theorem issetf 2770

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 2769	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2		issetf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	nfeq2 2351	. . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴
4		nfv 1542	. . 3 ⊢ Ⅎ𝑦 𝑥 = 𝐴
5		eqeq1 2203	. . 3 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴))
6	3, 4, 5	cbvex 1770	. 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 1506   ∈ wcel 2167  Ⅎwnfc 2326  Vcvv 2763

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  vtoclgf  2822  spcimgft  2840  spcimegft  2842  bj-vtoclgft  15421