Theorem axacprim 32459

Description: ax-ac 9679 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 26-Oct-2010.)

Assertion

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

Proof of Theorem axacprim

Step	Hyp	Ref	Expression
1		axacnd 9832	. 2 ⊢ ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))
2		df-an 388	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ↔ ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤))
3	2	albii 1782	. . . . . 6 ⊢ (∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ↔ ∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤))
4		anass 461	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ (𝑦 ∈ 𝑧 ∧ (𝑧 ∈ 𝑤 ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))))
5		annim 395	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑤 ∧ ¬ (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)) ↔ ¬ (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)))
6		pm4.63 389	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥) ↔ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))
7	6	anbi2i 613	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑤 ∧ ¬ (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑤 ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)))
8	5, 7	bitr3i 269	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)) ↔ (𝑧 ∈ 𝑤 ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)))
9	8	anbi2i 613	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑧 ∧ ¬ (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) ↔ (𝑦 ∈ 𝑧 ∧ (𝑧 ∈ 𝑤 ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))))
10		annim 395	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑧 ∧ ¬ (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) ↔ ¬ (𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))
11	4, 9, 10	3bitr2i 291	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ ¬ (𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))
12	11	exbii 1810	. . . . . . . . . . . 12 ⊢ (∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ ∃𝑤 ¬ (𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))
13		exnal 1789	. . . . . . . . . . . 12 ⊢ (∃𝑤 ¬ (𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) ↔ ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))
14	12, 13	bitri 267	. . . . . . . . . . 11 ⊢ (∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))
15	14	bibi1i 331	. . . . . . . . . 10 ⊢ ((∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤) ↔ (¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) ↔ 𝑦 = 𝑤))
16		dfbi1 205	. . . . . . . . . 10 ⊢ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) ↔ 𝑦 = 𝑤) ↔ ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))))
17	15, 16	bitri 267	. . . . . . . . 9 ⊢ ((∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤) ↔ ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))))
18	17	albii 1782	. . . . . . . 8 ⊢ (∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤) ↔ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))))
19	18	exbii 1810	. . . . . . 7 ⊢ (∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤) ↔ ∃𝑤∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))))
20		df-ex 1743	. . . . . . 7 ⊢ (∃𝑤∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))) ↔ ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))))
21	19, 20	bitri 267	. . . . . 6 ⊢ (∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤) ↔ ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))))
22	3, 21	imbi12i 343	. . . . 5 ⊢ ((∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)) ↔ (∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)))))))
23	22	2albii 1783	. . . 4 ⊢ (∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)) ↔ ∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)))))))
24	23	exbii 1810	. . 3 ⊢ (∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)) ↔ ∃𝑥∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)))))))
25		df-ex 1743	. . 3 ⊢ (∃𝑥∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)))))) ↔ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)))))))
26	24, 25	bitri 267	. 2 ⊢ (∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)) ↔ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)))))))
27	1, 26	mpbi 222	1 ⊢ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505  ∃wex 1742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-reg 8851  ax-ac 9679

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-eprel 5317  df-fr 5366

This theorem is referenced by: (None)