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Theorem issetf 3486
Description: A version of isset 3484 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 𝑥𝐴
Assertion
Ref Expression
issetf (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Proof of Theorem issetf
StepHypRef Expression
1 issetf.1 . 2 𝑥𝐴
2 issetft 3485 . 2 (𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))
31, 2ax-mp 5 1 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wex 1774  wcel 2099  wnfc 2879  Vcvv 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-v 3473
This theorem is referenced by:  vtoclgf  3555  spcimgft  3574  fineqvrep  34715
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