Axiom ax-regs 35639

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 35652, but this derivation relies on ax-inf2 9623 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 1812	. 2 wff ∃𝑥𝜑
4		vy	. . . . . . 7 setvar 𝑦
5	2, 4	weq 1995	. . . . . 6 wff 𝑥 = 𝑦
6	5, 1	wi 4	. . . . 5 wff (𝑥 = 𝑦 → 𝜑)
7	6, 2	wal 1568	. . . 4 wff ∀𝑥(𝑥 = 𝑦 → 𝜑)
8		vz	. . . . . . 7 setvar 𝑧
9	8, 4	wel 2146	. . . . . 6 wff 𝑧 ∈ 𝑦
10	2, 8	weq 1995	. . . . . . . . 9 wff 𝑥 = 𝑧
11	10, 1	wi 4	. . . . . . . 8 wff (𝑥 = 𝑧 → 𝜑)
12	11, 2	wal 1568	. . . . . . 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 1568	. . . 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 1812	. 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  35640  axregscl  35641