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Theorem issetf 3181
 Description: A version of isset 3180 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
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 isset 3180 . 2 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2 issetf.1 . . . 4 𝑥𝐴
32nfeq2 2766 . . 3 𝑥 𝑦 = 𝐴
4 nfv 1830 . . 3 𝑦 𝑥 = 𝐴
5 eqeq1 2614 . . 3 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
63, 4, 5cbvex 2260 . 2 (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
71, 6bitri 263 1 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Ⅎwnfc 2738  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175 This theorem is referenced by:  vtoclgf  3237  spcimgft  3257
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