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Theorem zfregcl 8382
 Description: The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.)
Assertion
Ref Expression
zfregcl (𝐴𝑉 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem zfregcl
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2677 . . . 4 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
21exbidv 1837 . . 3 (𝑧 = 𝐴 → (∃𝑥 𝑥𝑧 ↔ ∃𝑥 𝑥𝐴))
3 eleq2 2677 . . . . . 6 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
43notbid 307 . . . . 5 (𝑧 = 𝐴 → (¬ 𝑦𝑧 ↔ ¬ 𝑦𝐴))
54ralbidv 2969 . . . 4 (𝑧 = 𝐴 → (∀𝑦𝑥 ¬ 𝑦𝑧 ↔ ∀𝑦𝑥 ¬ 𝑦𝐴))
65rexeqbi1dv 3124 . . 3 (𝑧 = 𝐴 → (∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧 ↔ ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
72, 6imbi12d 333 . 2 (𝑧 = 𝐴 → ((∃𝑥 𝑥𝑧 → ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧) ↔ (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)))
8 nfre1 2988 . . 3 𝑥𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧
9 axreg2 8381 . . . 4 (𝑥𝑧 → ∃𝑥(𝑥𝑧 ∧ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧)))
10 df-ral 2901 . . . . . 6 (∀𝑦𝑥 ¬ 𝑦𝑧 ↔ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧))
1110rexbii 3023 . . . . 5 (∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧 ↔ ∃𝑥𝑧𝑦(𝑦𝑥 → ¬ 𝑦𝑧))
12 df-rex 2902 . . . . 5 (∃𝑥𝑧𝑦(𝑦𝑥 → ¬ 𝑦𝑧) ↔ ∃𝑥(𝑥𝑧 ∧ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧)))
1311, 12bitr2i 264 . . . 4 (∃𝑥(𝑥𝑧 ∧ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧)) ↔ ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧)
149, 13sylib 207 . . 3 (𝑥𝑧 → ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧)
158, 14exlimi 2073 . 2 (∃𝑥 𝑥𝑧 → ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧)
167, 15vtoclg 3239 1 (𝐴𝑉 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175 This theorem is referenced by:  zfreg  8383  elirrv  8387
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