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Theorem issetf 2744
Description: A version of isset 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 2743 . 2 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2 issetf.1 . . . 4 𝑥𝐴
32nfeq2 2331 . . 3 𝑥 𝑦 = 𝐴
4 nfv 1528 . . 3 𝑦 𝑥 = 𝐴
5 eqeq1 2184 . . 3 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
63, 4, 5cbvex 1756 . 2 (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
71, 6bitri 184 1 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353  wex 1492  wcel 2148  wnfc 2306  Vcvv 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  vtoclgf  2795  spcimgft  2813  spcimegft  2815  bj-vtoclgft  14447
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