Theorem wl-nfeqfb 32502

Description: Extend nfeqf 2289 to an equivalence. (Contributed by Wolf Lammen, 31-Jul-2019.)

Assertion

Ref	Expression
wl-nfeqfb	⊢ (Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))

Proof of Theorem wl-nfeqfb

Step	Hyp	Ref	Expression
1		nf5r 2052	. . . . 5 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
2	1	imp 444	. . . 4 ⊢ ((Ⅎ𝑥 𝑦 = 𝑧 ∧ 𝑦 = 𝑧) → ∀𝑥 𝑦 = 𝑧)
3		wl-aleq 32501	. . . . 5 ⊢ (∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))
4	3	simprbi 479	. . . 4 ⊢ (∀𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
5	2, 4	syl 17	. . 3 ⊢ ((Ⅎ𝑥 𝑦 = 𝑧 ∧ 𝑦 = 𝑧) → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
6		nfnt 1767	. . . . . 6 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → Ⅎ𝑥 ¬ 𝑦 = 𝑧)
7	6	nf5rd 2054	. . . . 5 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → (¬ 𝑦 = 𝑧 → ∀𝑥 ¬ 𝑦 = 𝑧))
8	7	imp 444	. . . 4 ⊢ ((Ⅎ𝑥 𝑦 = 𝑧 ∧ ¬ 𝑦 = 𝑧) → ∀𝑥 ¬ 𝑦 = 𝑧)
9		alnex 1697	. . . . . 6 ⊢ (∀𝑥 ¬ 𝑦 = 𝑧 ↔ ¬ ∃𝑥 𝑦 = 𝑧)
10		wl-exeq 32500	. . . . . 6 ⊢ (∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
11	9, 10	xchbinx 323	. . . . 5 ⊢ (∀𝑥 ¬ 𝑦 = 𝑧 ↔ ¬ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
12		3ioran 1049	. . . . 5 ⊢ (¬ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ↔ (¬ 𝑦 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧))
13	11, 12	sylbb 208	. . . 4 ⊢ (∀𝑥 ¬ 𝑦 = 𝑧 → (¬ 𝑦 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧))
14		3simpc 1053	. . . 4 ⊢ ((¬ 𝑦 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧))
15		pm5.21 899	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
16	8, 13, 14, 15	4syl 19	. . 3 ⊢ ((Ⅎ𝑥 𝑦 = 𝑧 ∧ ¬ 𝑦 = 𝑧) → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
17	5, 16	pm2.61dan 828	. 2 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
18		ax7 1930	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
19	18	al2imi 1733	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
20		nftht0 1709	. . . 4 ⊢ (∀𝑥 𝑦 = 𝑧 → Ⅎ𝑥 𝑦 = 𝑧)
21	19, 20	syl6 34	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑦 = 𝑧))
22		nfeqf 2289	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦 = 𝑧)
23	22	ex 449	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑦 = 𝑧))
24	21, 23	bija 369	. 2 ⊢ ((∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦 = 𝑧)
25	17, 24	impbii 198	1 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))

Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)