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Theorem wl-issetft 37563
Description: A closed form of issetf 3495. The proof here is a modification of a subproof in vtoclgft 3552, 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 3492 . 2 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2 nfv 1912 . . . . 5 𝑦𝑥𝐴
3 nfnfc1 2906 . . . . 5 𝑥𝑥𝐴
4 nfcvd 2904 . . . . . . 7 (𝑥𝐴𝑥𝑦)
5 id 22 . . . . . . 7 (𝑥𝐴𝑥𝐴)
64, 5nfeqd 2914 . . . . . 6 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
76nfnd 1856 . . . . 5 (𝑥𝐴 → Ⅎ𝑥 ¬ 𝑦 = 𝐴)
8 nfvd 1913 . . . . 5 (𝑥𝐴 → Ⅎ𝑦 ¬ 𝑥 = 𝐴)
9 eqeq1 2739 . . . . . . 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 1778 . . . 4 (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴)
14 alnex 1778 . . . 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 1535   = wceq 1537  wex 1776  wcel 2106  wnfc 2888  Vcvv 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480
This theorem is referenced by: (None)
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