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Theorem issetf 3071
Description: A version of isset 3070 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 3070 . 2 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2 issetf.1 . . . 4 𝑥𝐴
32nfeq2 2661 . . 3 𝑥 𝑦 = 𝐴
4 nfv 1792 . . 3 𝑦 𝑥 = 𝐴
5 eqeq1 2509 . . 3 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
63, 4, 5cbvex 2162 . 2 (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
71, 6bitri 259 1 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 191   = wceq 1468  wex 1692  wcel 1937  wnfc 2633  Vcvv 3066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485
This theorem depends on definitions:  df-bi 192  df-an 380  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-v 3068
This theorem is referenced by:  vtoclgf  3126  spcimgft  3146
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