Theorem axextprim 33309

Description: ax-ext 2708 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 10170	. 2 ⊢ ∃𝑥((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧)
2		dfbi2 478	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) ↔ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) ∧ (𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦)))
3	2	imbi1i 353	. . . . 5 ⊢ (((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) ↔ (((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) ∧ (𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦)) → 𝑦 = 𝑧))
4		impexp 454	. . . . 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 278	. . . 4 ⊢ (((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) ↔ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧)))
6	5	exbii 1855	. . 3 ⊢ (∃𝑥((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) ↔ ∃𝑥((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧)))
7		df-ex 1788	. . 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 278	. 2 ⊢ (∃𝑥((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) ↔ ¬ ∀𝑥 ¬ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧)))
9	1, 8	mpbi 233	1 ⊢ ¬ ∀𝑥 ¬ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-13 2371  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-clel 2809  df-nfc 2879

This theorem is referenced by: (None)