Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-equsb3 Structured version   Visualization version   GIF version

Theorem wl-equsb3 34674
Description: equsb3 2100 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.)
Assertion
Ref Expression
wl-equsb3 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧))

Proof of Theorem wl-equsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfna1 2147 . . . 4 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
2 nfeqf2 2388 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 = 𝑧)
3 equequ1 2023 . . . . 5 (𝑦 = 𝑤 → (𝑦 = 𝑧𝑤 = 𝑧))
43a1i 11 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑤 → (𝑦 = 𝑧𝑤 = 𝑧)))
51, 2, 4sbied 2541 . . 3 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑤 / 𝑦]𝑦 = 𝑧𝑤 = 𝑧))
65sbbidv 2075 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧))
7 sbco2vv 2099 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
8 equsb3 2100 . 2 ([𝑥 / 𝑤]𝑤 = 𝑧𝑥 = 𝑧)
96, 7, 83bitr3g 314 1 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wal 1526  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167  ax-13 2383
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator