Theorem issetf 3489

Description: A version of isset 3488 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 3488	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2		issetf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	nfeq2 2921	. . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐴
4		nfv 1918	. . 3 ⊢ Ⅎ𝑦 𝑥 = 𝐴
5		eqeq1 2737	. . 3 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴))
6	3, 4, 5	cbvexv1 2339	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
7	1, 6	bitri 275	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Syntax hints:   ↔ wb 205   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Ⅎwnfc 2884  Vcvv 3475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3477

This theorem is referenced by:  vtoclgf  3555  spcimgft  3578  fineqvrep  34095