Theorem zfregclOLD 9476

Description: Obsolete version of zfregcl 9475 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 2818	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐴))
2	1	exbidv 1922	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 𝑥 ∈ 𝑧 ↔ ∃𝑥 𝑥 ∈ 𝐴))
3		eleq2 2818	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝐴))
4	3	notbid 318	. . . . 5 ⊢ (𝑧 = 𝐴 → (¬ 𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝐴))
5	4	ralbidv 3153	. . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))
6	5	rexeqbi1dv 3303	. . 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 3255	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧
9		axreg2 9474	. . . 4 ⊢ (𝑥 ∈ 𝑧 → ∃𝑥(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧)))
10		df-ral 3046	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧))
11	10	rexbii 3077	. . . . 5 ⊢ (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑧 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧))
12		df-rex 3055	. . . . 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 2219	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧)
16	7, 15	vtoclg 3507	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2110  ∀wral 3045  ∃wrex 3054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-12 2179  ax-ext 2702  ax-reg 9473

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055

This theorem is referenced by: (None)