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Theorem wl-equsb3 34233
 Description: equsb3 2045 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 2089 . . . 4 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
2 nfeqf2 2306 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 = 𝑧)
3 equequ1 1982 . . . . 5 (𝑦 = 𝑤 → (𝑦 = 𝑧𝑤 = 𝑧))
43a1i 11 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑤 → (𝑦 = 𝑧𝑤 = 𝑧)))
51, 2, 4sbied 2469 . . 3 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑤 / 𝑦]𝑦 = 𝑧𝑤 = 𝑧))
65sbbidv 2030 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧))
7 sbco2vv 2044 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
8 equsb3 2045 . 2 ([𝑥 / 𝑤]𝑤 = 𝑧𝑥 = 𝑧)
96, 7, 83bitr3g 305 1 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198  ∀wal 1505  [wsb 2015 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-10 2079  ax-12 2106  ax-13 2301 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-nf 1747  df-sb 2016 This theorem is referenced by: (None)
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