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Theorem wl-equsb3 32966
Description: equsb3 2431 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 nfv 1840 . . 3 𝑤 ¬ ∀𝑦 𝑦 = 𝑧
2 nfna1 2026 . . . 4 𝑦 ¬ ∀𝑦 𝑦 = 𝑧
3 nfeqf2 2296 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑤 = 𝑧)
4 equequ1 1949 . . . . 5 (𝑦 = 𝑤 → (𝑦 = 𝑧𝑤 = 𝑧))
54a1i 11 . . . 4 (¬ ∀𝑦 𝑦 = 𝑧 → (𝑦 = 𝑤 → (𝑦 = 𝑧𝑤 = 𝑧)))
62, 3, 5sbied 2408 . . 3 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑤 / 𝑦]𝑦 = 𝑧𝑤 = 𝑧))
71, 6sbbid 2402 . 2 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧))
8 sbcom3 2410 . . 3 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤][𝑥 / 𝑦]𝑦 = 𝑧)
9 nfv 1840 . . . 4 𝑤[𝑥 / 𝑦]𝑦 = 𝑧
109sbf 2379 . . 3 ([𝑥 / 𝑤][𝑥 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
118, 10bitri 264 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
12 equsb3 2431 . 2 ([𝑥 / 𝑤]𝑤 = 𝑧𝑥 = 𝑧)
137, 11, 123bitr3g 302 1 (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1478  [wsb 1877
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-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by: (None)
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