Theorem axextprim

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

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

Proof

Step	Hyp	Ref	Expression
1		axextnd	⊢ ∃ 𝑥 ( ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) → 𝑦 = 𝑧 )
2		dfbi2	⊢ ( ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) ↔ ( ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) ∧ ( 𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦 ) ) )
3	2	imbi1i	⊢ ( ( ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) → 𝑦 = 𝑧 ) ↔ ( ( ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) ∧ ( 𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦 ) ) → 𝑦 = 𝑧 ) )
4		impexp	⊢ ( ( ( ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) ∧ ( 𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦 ) ) → 𝑦 = 𝑧 ) ↔ ( ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) → ( ( 𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦 ) → 𝑦 = 𝑧 ) ) )
5	3 4	bitri	⊢ ( ( ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) → 𝑦 = 𝑧 ) ↔ ( ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) → ( ( 𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦 ) → 𝑦 = 𝑧 ) ) )
6	5	exbii	⊢ ( ∃ 𝑥 ( ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) → 𝑦 = 𝑧 ) ↔ ∃ 𝑥 ( ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) → ( ( 𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦 ) → 𝑦 = 𝑧 ) ) )
7		df-ex	⊢ ( ∃ 𝑥 ( ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) → ( ( 𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦 ) → 𝑦 = 𝑧 ) ) ↔ ¬ ∀ 𝑥 ¬ ( ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) → ( ( 𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦 ) → 𝑦 = 𝑧 ) ) )
8	6 7	bitri	⊢ ( ∃ 𝑥 ( ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) → 𝑦 = 𝑧 ) ↔ ¬ ∀ 𝑥 ¬ ( ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) → ( ( 𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦 ) → 𝑦 = 𝑧 ) ) )
9	1 8	mpbi	⊢ ¬ ∀ 𝑥 ¬ ( ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) → ( ( 𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦 ) → 𝑦 = 𝑧 ) )