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Theorem wl-issetft 37584
Description: A closed form of issetf 3496. The proof here is a modification of a subproof in vtoclgft 3551, where it could be used to shorten the proof. (Contributed by Wolf Lammen, 25-Jan-2025.)
Assertion
Ref Expression
wl-issetft (𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))

Proof of Theorem wl-issetft
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 isset 3493 . 2 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2 nfv 1913 . . . . 5 𝑦𝑥𝐴
3 nfnfc1 2907 . . . . 5 𝑥𝑥𝐴
4 nfcvd 2905 . . . . . . 7 (𝑥𝐴𝑥𝑦)
5 id 22 . . . . . . 7 (𝑥𝐴𝑥𝐴)
64, 5nfeqd 2915 . . . . . 6 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
76nfnd 1857 . . . . 5 (𝑥𝐴 → Ⅎ𝑥 ¬ 𝑦 = 𝐴)
8 nfvd 1914 . . . . 5 (𝑥𝐴 → Ⅎ𝑦 ¬ 𝑥 = 𝐴)
9 eqeq1 2740 . . . . . . 7 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
109notbid 318 . . . . . 6 (𝑦 = 𝑥 → (¬ 𝑦 = 𝐴 ↔ ¬ 𝑥 = 𝐴))
1110a1i 11 . . . . 5 (𝑥𝐴 → (𝑦 = 𝑥 → (¬ 𝑦 = 𝐴 ↔ ¬ 𝑥 = 𝐴)))
122, 3, 7, 8, 11cbv2w 2338 . . . 4 (𝑥𝐴 → (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ∀𝑥 ¬ 𝑥 = 𝐴))
13 alnex 1780 . . . 4 (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴)
14 alnex 1780 . . . 4 (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
1512, 13, 143bitr3g 313 . . 3 (𝑥𝐴 → (¬ ∃𝑦 𝑦 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴))
1615con4bid 317 . 2 (𝑥𝐴 → (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
171, 16bitrid 283 1 (𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wal 1537   = wceq 1539  wex 1778  wcel 2107  wnfc 2889  Vcvv 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-v 3481
This theorem is referenced by: (None)
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