Theorem axreg 35640

Description: Derivation of ax-reg 9567 from ax-regs 35639 and Tarski's FOL axiom schemes. This demonstrates the sense in which ax-regs 35639 is a stronger version of ax-reg 9567. (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 35639	. 2 ⊢ (∃𝑤 𝑤 ∈ 𝑥 → ∃𝑦(∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑥) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥))))
2		elequ1 2152	. . 3 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
3	2	cbvexvw 2070	. 2 ⊢ (∃𝑤 𝑤 ∈ 𝑥 ↔ ∃𝑦 𝑦 ∈ 𝑥)
4	2	equsalvw 2037	. . . 4 ⊢ (∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑥) ↔ 𝑦 ∈ 𝑥)
5		elequ1 2152	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
6	5	equsalvw 2037	. . . . . . 7 ⊢ (∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥) ↔ 𝑧 ∈ 𝑥)
7	6	notbii 323	. . . . . 6 ⊢ (¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥) ↔ ¬ 𝑧 ∈ 𝑥)
8	7	imbi2i 339	. . . . 5 ⊢ ((𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))
9	8	albii 1852	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥)) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))
10	4, 9	anbi12i 640	. . 3 ⊢ ((∀𝑤(𝑤 = 𝑦 → 𝑤 ∈ 𝑥) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑤(𝑤 = 𝑧 → 𝑤 ∈ 𝑥))) ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
11	10	exbii 1881	. 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 294	1 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-regs 35639

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)