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Theorem wl-equsb3 36724
Description: equsb3 2099 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 2374 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 = 𝑧)
3 equequ1 2026 . . . . 5 (𝑦 = 𝑤 → (𝑦 = 𝑧𝑤 = 𝑧))
43a1i 11 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑤 → (𝑦 = 𝑧𝑤 = 𝑧)))
51, 2, 4sbied 2500 . . 3 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑤 / 𝑦]𝑦 = 𝑧𝑤 = 𝑧))
65sbbidv 2080 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧))
7 sbco2vv 2098 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
8 equsb3 2099 . 2 ([𝑥 / 𝑤]𝑤 = 𝑧𝑥 = 𝑧)
96, 7, 83bitr3g 312 1 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1537  [wsb 2065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-12 2169  ax-13 2369
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ex 1780  df-nf 1784  df-sb 2066
This theorem is referenced by: (None)
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