Theorem zfregclOLD 9504

Description: Obsolete version of zfregcl 9503 as of 31-Dec-2025. (Contributed by NM, 5-Aug-1994.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
zfregclOLD	⊢ (𝐴 ∈ 𝑉 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem zfregclOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2826	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐴))
2	1	exbidv 1923	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 𝑥 ∈ 𝑧 ↔ ∃𝑥 𝑥 ∈ 𝐴))
3		eleq2 2826	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝐴))
4	3	notbid 318	. . . . 5 ⊢ (𝑧 = 𝐴 → (¬ 𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝐴))
5	4	ralbidv 3160	. . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))
6	5	rexeqbi1dv 3310	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))
7	2, 6	imbi12d 344	. 2 ⊢ (𝑧 = 𝐴 → ((∃𝑥 𝑥 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧) ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴)))
8		nfre1 3262	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧
9		axreg2 9502	. . . 4 ⊢ (𝑥 ∈ 𝑧 → ∃𝑥(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧)))
10		df-ral 3053	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧))
11	10	rexbii 3084	. . . . 5 ⊢ (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑧 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧))
12		df-rex 3062	. . . . 5 ⊢ (∃𝑥 ∈ 𝑧 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧) ↔ ∃𝑥(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧)))
13	11, 12	bitr2i 276	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧)) ↔ ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧)
14	9, 13	sylib 218	. . 3 ⊢ (𝑥 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧)
15	8, 14	exlimi 2225	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧)
16	7, 15	vtoclg 3512	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-reg 9501

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062

This theorem is referenced by: (None)