Theorem axreg

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

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

Proof

Step	Hyp	Ref	Expression
1		ax-regs	⊢ ( ∃ 𝑤 𝑤 ∈ 𝑥 → ∃ 𝑦 ( ∀ 𝑤 ( 𝑤 = 𝑦 → 𝑤 ∈ 𝑥 ) ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ¬ ∀ 𝑤 ( 𝑤 = 𝑧 → 𝑤 ∈ 𝑥 ) ) ) )
2		elequ1	⊢ ( 𝑤 = 𝑦 → ( 𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥 ) )
3	2	cbvexvw	⊢ ( ∃ 𝑤 𝑤 ∈ 𝑥 ↔ ∃ 𝑦 𝑦 ∈ 𝑥 )
4	2	equsalvw	⊢ ( ∀ 𝑤 ( 𝑤 = 𝑦 → 𝑤 ∈ 𝑥 ) ↔ 𝑦 ∈ 𝑥 )
5		elequ1	⊢ ( 𝑤 = 𝑧 → ( 𝑤 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥 ) )
6	5	equsalvw	⊢ ( ∀ 𝑤 ( 𝑤 = 𝑧 → 𝑤 ∈ 𝑥 ) ↔ 𝑧 ∈ 𝑥 )
7	6	notbii	⊢ ( ¬ ∀ 𝑤 ( 𝑤 = 𝑧 → 𝑤 ∈ 𝑥 ) ↔ ¬ 𝑧 ∈ 𝑥 )
8	7	imbi2i	⊢ ( ( 𝑧 ∈ 𝑦 → ¬ ∀ 𝑤 ( 𝑤 = 𝑧 → 𝑤 ∈ 𝑥 ) ) ↔ ( 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥 ) )
9	8	albii	⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ¬ ∀ 𝑤 ( 𝑤 = 𝑧 → 𝑤 ∈ 𝑥 ) ) ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥 ) )
10	4 9	anbi12i	⊢ ( ( ∀ 𝑤 ( 𝑤 = 𝑦 → 𝑤 ∈ 𝑥 ) ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ¬ ∀ 𝑤 ( 𝑤 = 𝑧 → 𝑤 ∈ 𝑥 ) ) ) ↔ ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥 ) ) )
11	10	exbii	⊢ ( ∃ 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	⊢ ( ∃ 𝑦 𝑦 ∈ 𝑥 → ∃ 𝑦 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥 ) ) )