Theorem axextprim 32175

Description: ax-ext 2754 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)

Assertion

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

Proof of Theorem axextprim

Step	Hyp	Ref	Expression
1		axextnd 9748	. 2 ⊢ ∃𝑥((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧)
2		dfbi2 468	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) ↔ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) ∧ (𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦)))
3	2	imbi1i 341	. . . . 5 ⊢ (((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) ↔ (((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) ∧ (𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦)) → 𝑦 = 𝑧))
4		impexp 443	. . . . 5 ⊢ ((((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) ∧ (𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦)) → 𝑦 = 𝑧) ↔ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧)))
5	3, 4	bitri 267	. . . 4 ⊢ (((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) ↔ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧)))
6	5	exbii 1892	. . 3 ⊢ (∃𝑥((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) ↔ ∃𝑥((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧)))
7		df-ex 1824	. . 3 ⊢ (∃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 267	. 2 ⊢ (∃𝑥((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) ↔ ¬ ∀𝑥 ¬ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧)))
9	1, 8	mpbi 222	1 ⊢ ¬ ∀𝑥 ¬ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599  ∃wex 1823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-nfc 2921

This theorem is referenced by: (None)