Theorem issetf 2779

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 2778	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2		issetf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	nfeq2 2360	. . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴
4		nfv 1551	. . 3 ⊢ Ⅎ𝑦 𝑥 = 𝐴
5		eqeq1 2212	. . 3 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴))
6	3, 4, 5	cbvex 1779	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
7	1, 6	bitri 184	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1373  ∃wex 1515   ∈ wcel 2176  Ⅎwnfc 2335  Vcvv 2772

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774

This theorem is referenced by:  vtoclgf  2831  spcimgft  2849  spcimegft  2851  bj-vtoclgft  15715