Theorem axregprim

Description: ax-reg without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010)

	Ref	Expression
Assertion	axregprim	⊢ ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑥 ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		axregnd	⊢ ( 𝑥 ∈ 𝑦 → ∃ 𝑥 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦 ) ) )
2		df-an	⊢ ( ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦 ) ) ↔ ¬ ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦 ) ) )
3	2	exbii	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦 ) ) ↔ ∃ 𝑥 ¬ ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦 ) ) )
4		exnal	⊢ ( ∃ 𝑥 ¬ ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦 ) ) ↔ ¬ ∀ 𝑥 ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦 ) ) )
5	3 4	bitri	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦 ) ) ↔ ¬ ∀ 𝑥 ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦 ) ) )
6	1 5	sylib	⊢ ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑥 ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦 ) ) )