Axiom ax-regs 35836

Description: A strong version of the Axiom of Regularity. It states that if there exists a set with property 𝜑, then there must exist a set with property 𝜑 such that none of its elements have property 𝜑. This axiom can be derived from the axioms of ZF set theory as shown in axregs 35849, but this derivation relies on ax-inf2 9642 and is thus not possible in a finitist context. (Contributed by BTernaryTau, 29-Dec-2025.)

Assertion

Ref	Expression
ax-regs	⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

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

Allowed substitution hint:   𝜑(𝑥)

Detailed syntax breakdown of Axiom ax-regs

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3	1, 2	wex 1811	. 2 wff ∃𝑥𝜑
4		vy	. . . . . . 7 setvar 𝑦
5	2, 4	weq 1994	. . . . . 6 wff 𝑥 = 𝑦
6	5, 1	wi 4	. . . . 5 wff (𝑥 = 𝑦 → 𝜑)
7	6, 2	wal 1567	. . . 4 wff ∀𝑥(𝑥 = 𝑦 → 𝜑)
8		vz	. . . . . . 7 setvar 𝑧
9	8, 4	wel 2145	. . . . . 6 wff 𝑧 ∈ 𝑦
10	2, 8	weq 1994	. . . . . . . . 9 wff 𝑥 = 𝑧
11	10, 1	wi 4	. . . . . . . 8 wff (𝑥 = 𝑧 → 𝜑)
12	11, 2	wal 1567	. . . . . . 7 wff ∀𝑥(𝑥 = 𝑧 → 𝜑)
13	12	wn 3	. . . . . 6 wff ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)
14	9, 13	wi 4	. . . . 5 wff (𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))
15	14, 8	wal 1567	. . . 4 wff ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))
16	7, 15	wa 401	. . 3 wff (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
17	16, 4	wex 1811	. 2 wff ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
18	3, 17	wi 4	1 wff (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

This axiom is used by:  axreg  35837  axregscl  35838