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Theorem wl-issetft 36063
Description: A closed form of issetf 3462. The proof here is a modification of a subproof in vtoclgft 3512, 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 3461 . 2 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2 nfv 1918 . . . . 5 𝑦𝑥𝐴
3 nfnfc1 2911 . . . . 5 𝑥𝑥𝐴
4 nfcvd 2909 . . . . . . 7 (𝑥𝐴𝑥𝑦)
5 id 22 . . . . . . 7 (𝑥𝐴𝑥𝐴)
64, 5nfeqd 2918 . . . . . 6 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
76nfnd 1862 . . . . 5 (𝑥𝐴 → Ⅎ𝑥 ¬ 𝑦 = 𝐴)
8 nfvd 1919 . . . . 5 (𝑥𝐴 → Ⅎ𝑦 ¬ 𝑥 = 𝐴)
9 eqeq1 2741 . . . . . . 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 2334 . . . 4 (𝑥𝐴 → (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ∀𝑥 ¬ 𝑥 = 𝐴))
13 alnex 1784 . . . 4 (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴)
14 alnex 1784 . . . 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 205  wal 1540   = wceq 1542  wex 1782  wcel 2107  wnfc 2888  Vcvv 3448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-v 3450
This theorem is referenced by: (None)
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