Theorem axregscl 35316

Description: A version of ax-regs 35314 with a class variable instead of a wff variable. Axiom D in Gödel, The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (1940), p. 6. (Contributed by BTernaryTau, 30-Dec-2025.)

Assertion

Ref	Expression
axregscl	⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴,𝑧

Proof of Theorem axregscl

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2823	. . 3 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
2	1	cbvexvw 2044	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑤 𝑤 ∈ 𝐴)
3		ax-regs 35314	. . 3 ⊢ (∃𝑤 𝑤 ∈ 𝐴 → ∃𝑦(∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝐴) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝐴))))
4		eleq1w 2823	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
5	4	equsalvw 2011	. . . . 5 ⊢ (∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴)
6		eleq1w 2823	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
7	6	equsalvw 2011	. . . . . . . 8 ⊢ (∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝐴) ↔ 𝑧 ∈ 𝐴)
8	7	notbii 321	. . . . . . 7 ⊢ (¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝐴) ↔ ¬ 𝑧 ∈ 𝐴)
9	8	imbi2i 337	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝐴)) ↔ (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴))
10	9	albii 1826	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝐴)) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴))
11	5, 10	anbi12i 634	. . . 4 ⊢ ((∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝐴) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝐴))) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)))
12	11	exbii 1855	. . 3 ⊢ (∃𝑦(∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝐴) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝐴))) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)))
13	3, 12	sylib 219	. 2 ⊢ (∃𝑤 𝑤 ∈ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)))
14	2, 13	sylbi 218	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1545  ∃wex 1786   ∈ wcel 2119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-regs 35314

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-clel 2815

This theorem is referenced by:  axregszf  35317