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Theorem zfregcl 9047
 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 2906 . . . 4 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
21exbidv 1915 . . 3 (𝑧 = 𝐴 → (∃𝑥 𝑥𝑧 ↔ ∃𝑥 𝑥𝐴))
3 eleq2 2906 . . . . . 6 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
43notbid 319 . . . . 5 (𝑧 = 𝐴 → (¬ 𝑦𝑧 ↔ ¬ 𝑦𝐴))
54ralbidv 3202 . . . 4 (𝑧 = 𝐴 → (∀𝑦𝑥 ¬ 𝑦𝑧 ↔ ∀𝑦𝑥 ¬ 𝑦𝐴))
65rexeqbi1dv 3410 . . 3 (𝑧 = 𝐴 → (∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧 ↔ ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
72, 6imbi12d 346 . 2 (𝑧 = 𝐴 → ((∃𝑥 𝑥𝑧 → ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧) ↔ (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)))
8 nfre1 3311 . . 3 𝑥𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧
9 axreg2 9046 . . . 4 (𝑥𝑧 → ∃𝑥(𝑥𝑧 ∧ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧)))
10 df-ral 3148 . . . . . 6 (∀𝑦𝑥 ¬ 𝑦𝑧 ↔ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧))
1110rexbii 3252 . . . . 5 (∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧 ↔ ∃𝑥𝑧𝑦(𝑦𝑥 → ¬ 𝑦𝑧))
12 df-rex 3149 . . . . 5 (∃𝑥𝑧𝑦(𝑦𝑥 → ¬ 𝑦𝑧) ↔ ∃𝑥(𝑥𝑧 ∧ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧)))
1311, 12bitr2i 277 . . . 4 (∃𝑥(𝑥𝑧 ∧ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧)) ↔ ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧)
149, 13sylib 219 . . 3 (𝑥𝑧 → ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧)
158, 14exlimi 2210 . 2 (∃𝑥 𝑥𝑧 → ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧)
167, 15vtoclg 3573 1 (𝐴𝑉 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1528   = wceq 1530  ∃wex 1773   ∈ wcel 2107  ∀wral 3143  ∃wrex 3144 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2169  ax-ext 2798  ax-reg 9045 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778  df-cleq 2819  df-clel 2898  df-ral 3148  df-rex 3149 This theorem is referenced by:  zfreg  9048  elirrv  9049
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