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Theorem wl-nfeqfb 32982
Description: Extend nfeqf 2300 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
StepHypRef Expression
1 nf5r 2062 . . . . 5 (Ⅎ𝑥 𝑦 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
21imp 445 . . . 4 ((Ⅎ𝑥 𝑦 = 𝑧𝑦 = 𝑧) → ∀𝑥 𝑦 = 𝑧)
3 wl-aleq 32981 . . . . 5 (∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))
43simprbi 480 . . . 4 (∀𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
52, 4syl 17 . . 3 ((Ⅎ𝑥 𝑦 = 𝑧𝑦 = 𝑧) → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
6 nfnt 1779 . . . . . 6 (Ⅎ𝑥 𝑦 = 𝑧 → Ⅎ𝑥 ¬ 𝑦 = 𝑧)
76nf5rd 2064 . . . . 5 (Ⅎ𝑥 𝑦 = 𝑧 → (¬ 𝑦 = 𝑧 → ∀𝑥 ¬ 𝑦 = 𝑧))
87imp 445 . . . 4 ((Ⅎ𝑥 𝑦 = 𝑧 ∧ ¬ 𝑦 = 𝑧) → ∀𝑥 ¬ 𝑦 = 𝑧)
9 alnex 1703 . . . . . 6 (∀𝑥 ¬ 𝑦 = 𝑧 ↔ ¬ ∃𝑥 𝑦 = 𝑧)
10 wl-exeq 32980 . . . . . 6 (∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
119, 10xchbinx 324 . . . . 5 (∀𝑥 ¬ 𝑦 = 𝑧 ↔ ¬ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
12 3ioran 1054 . . . . 5 (¬ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ↔ (¬ 𝑦 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧))
1311, 12sylbb 209 . . . 4 (∀𝑥 ¬ 𝑦 = 𝑧 → (¬ 𝑦 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧))
14 3simpc 1058 . . . 4 ((¬ 𝑦 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧))
15 pm5.21 902 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
168, 13, 14, 154syl 19 . . 3 ((Ⅎ𝑥 𝑦 = 𝑧 ∧ ¬ 𝑦 = 𝑧) → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
175, 16pm2.61dan 831 . 2 (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
18 ax7 1940 . . . . 5 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
1918al2imi 1740 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
20 nftht 1715 . . . 4 (∀𝑥 𝑦 = 𝑧 → Ⅎ𝑥 𝑦 = 𝑧)
2119, 20syl6 35 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑦 = 𝑧))
22 nfeqf 2300 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦 = 𝑧)
2322ex 450 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑦 = 𝑧))
2421, 23bija 370 . 2 ((∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦 = 𝑧)
2517, 24impbii 199 1 (Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384  w3o 1035  w3a 1036  wal 1478  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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