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Theorem zfregclOLD 8446
 Description: Obsolete version of zfregcl 8444 as of 28-Apr-2021. (Contributed by NM, 5-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
zfregclOLD.1 𝐴 ∈ V
Assertion
Ref Expression
zfregclOLD (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem zfregclOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 zfregclOLD.1 . 2 𝐴 ∈ V
2 eleq2 2693 . . . 4 (𝑧 = 𝐴 → (𝑥𝑧𝑥𝐴))
32exbidv 1852 . . 3 (𝑧 = 𝐴 → (∃𝑥 𝑥𝑧 ↔ ∃𝑥 𝑥𝐴))
4 eleq2 2693 . . . . . 6 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
54notbid 308 . . . . 5 (𝑧 = 𝐴 → (¬ 𝑦𝑧 ↔ ¬ 𝑦𝐴))
65ralbidv 2985 . . . 4 (𝑧 = 𝐴 → (∀𝑦𝑥 ¬ 𝑦𝑧 ↔ ∀𝑦𝑥 ¬ 𝑦𝐴))
76rexeqbi1dv 3141 . . 3 (𝑧 = 𝐴 → (∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧 ↔ ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
83, 7imbi12d 334 . 2 (𝑧 = 𝐴 → ((∃𝑥 𝑥𝑧 → ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧) ↔ (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)))
9 nfre1 3004 . . 3 𝑥𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧
10 axreg2 8443 . . . 4 (𝑥𝑧 → ∃𝑥(𝑥𝑧 ∧ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧)))
11 df-ral 2917 . . . . . 6 (∀𝑦𝑥 ¬ 𝑦𝑧 ↔ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧))
1211rexbii 3039 . . . . 5 (∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧 ↔ ∃𝑥𝑧𝑦(𝑦𝑥 → ¬ 𝑦𝑧))
13 df-rex 2918 . . . . 5 (∃𝑥𝑧𝑦(𝑦𝑥 → ¬ 𝑦𝑧) ↔ ∃𝑥(𝑥𝑧 ∧ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧)))
1412, 13bitr2i 265 . . . 4 (∃𝑥(𝑥𝑧 ∧ ∀𝑦(𝑦𝑥 → ¬ 𝑦𝑧)) ↔ ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧)
1510, 14sylib 208 . . 3 (𝑥𝑧 → ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧)
169, 15exlimi 2089 . 2 (∃𝑥 𝑥𝑧 → ∃𝑥𝑧𝑦𝑥 ¬ 𝑦𝑧)
171, 8, 16vtocl 3250 1 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1478   = wceq 1480  ∃wex 1701   ∈ wcel 1992  ∀wral 2912  ∃wrex 2913  Vcvv 3191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-ext 2606  ax-reg 8442 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-v 3193 This theorem is referenced by:  zfregOLD  8447  zfreg2OLD  8448
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