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Theorem zfregcl 8540
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 2719 . . . 4 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
21exbidv 1890 . . 3 (𝑧 = 𝐴 → (∃𝑥 𝑥𝑧 ↔ ∃𝑥 𝑥𝐴))
3 eleq2 2719 . . . . . 6 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
43notbid 307 . . . . 5 (𝑧 = 𝐴 → (¬ 𝑦𝑧 ↔ ¬ 𝑦𝐴))
54ralbidv 3015 . . . 4 (𝑧 = 𝐴 → (∀𝑦𝑥 ¬ 𝑦𝑧 ↔ ∀𝑦𝑥 ¬ 𝑦𝐴))
65rexeqbi1dv 3177 . . 3 (𝑧 = 𝐴 → (∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧 ↔ ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
72, 6imbi12d 333 . 2 (𝑧 = 𝐴 → ((∃𝑥 𝑥𝑧 → ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧) ↔ (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)))
8 nfre1 3034 . . 3 𝑥𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧
9 axreg2 8539 . . . 4 (𝑥𝑧 → ∃𝑥(𝑥𝑧 ∧ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧)))
10 df-ral 2946 . . . . . 6 (∀𝑦𝑥 ¬ 𝑦𝑧 ↔ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧))
1110rexbii 3070 . . . . 5 (∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧 ↔ ∃𝑥𝑧𝑦(𝑦𝑥 → ¬ 𝑦𝑧))
12 df-rex 2947 . . . . 5 (∃𝑥𝑧𝑦(𝑦𝑥 → ¬ 𝑦𝑧) ↔ ∃𝑥(𝑥𝑧 ∧ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧)))
1311, 12bitr2i 265 . . . 4 (∃𝑥(𝑥𝑧 ∧ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧)) ↔ ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧)
149, 13sylib 208 . . 3 (𝑥𝑧 → ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧)
158, 14exlimi 2124 . 2 (∃𝑥 𝑥𝑧 → ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧)
167, 15vtoclg 3297 1 (𝐴𝑉 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-ext 2631  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233
This theorem is referenced by:  zfreg  8541  elirrv  8542
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