Theorem axreg 35427

Description: Derivation of ax-reg 9538 from ax-regs 35426 and Tarski's FOL axiom schemes. This demonstrates the sense in which ax-regs 35426 is a stronger version of ax-reg 9538. (Contributed by BTernaryTau, 30-Dec-2025.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axreg

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-regs 35426	. 2 ⊢ (∃𝑤 𝑤 ∈ 𝑥 → ∃𝑦(∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑥) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥))))
2		elequ1 2150	. . 3 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
3	2	cbvexvw 2058	. 2 ⊢ (∃𝑤 𝑤 ∈ 𝑥 ↔ ∃𝑦 𝑦 ∈ 𝑥)
4	2	equsalvw 2025	. . . 4 ⊢ (∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑥) ↔ 𝑦 ∈ 𝑥)
5		elequ1 2150	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
6	5	equsalvw 2025	. . . . . . 7 ⊢ (∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥) ↔ 𝑧 ∈ 𝑥)
7	6	notbii 322	. . . . . 6 ⊢ (¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥) ↔ ¬ 𝑧 ∈ 𝑥)
8	7	imbi2i 338	. . . . 5 ⊢ ((𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))
9	8	albii 1840	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥)) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))
10	4, 9	anbi12i 637	. . 3 ⊢ ((∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑥) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥))) ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
11	10	exbii 1869	. 2 ⊢ (∃𝑦(∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑥) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥))) ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
12	1, 3, 11	3imtr3i 293	1 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1559  ∃wex 1800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-regs 35426

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801

This theorem is referenced by: (None)