Theorem issetf 3473

Description: A version of isset 3470 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

Step	Hyp	Ref	Expression
1		issetf.1	. 2 ⊢ Ⅎ𝑥𝐴
2		issetft 3472	. 2 ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Syntax hints:   ↔ wb 208   = wceq 1562  ∃wex 1801   ∈ wcel 2144  Ⅎwnfc 2911  Vcvv 3456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-v 3458

This theorem is referenced by:  spcimgfi1OLD  3518  vtoclgf  3536  fineqvrep  35414