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Theorem wl-equsb3 34956
Description: equsb3 2107 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 2154 . . . 4 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
2 nfeqf2 2387 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 = 𝑧)
3 equequ1 2032 . . . . 5 (𝑦 = 𝑤 → (𝑦 = 𝑧𝑤 = 𝑧))
43a1i 11 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑤 → (𝑦 = 𝑧𝑤 = 𝑧)))
51, 2, 4sbied 2525 . . 3 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑤 / 𝑦]𝑦 = 𝑧𝑤 = 𝑧))
65sbbidv 2084 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧))
7 sbco2vv 2106 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
8 equsb3 2107 . 2 ([𝑥 / 𝑤]𝑤 = 𝑧𝑥 = 𝑧)
96, 7, 83bitr3g 316 1 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wal 1536  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070
This theorem is referenced by: (None)
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