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Theorem wl-equsb4 33829
Description: Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable constraint. (Contributed by Wolf Lammen, 26-Jun-2019.)
Assertion
Ref Expression
wl-equsb4 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧))

Proof of Theorem wl-equsb4
StepHypRef Expression
1 nfeqf 2388 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦 = 𝑧)
21ex 402 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑦 = 𝑧))
3 sbft 2496 . . 3 (Ⅎ𝑥 𝑦 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧))
42, 3syl6com 37 . 2 (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧)))
5 sbequ12r 2279 . . . 4 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧))
65equcoms 2119 . . 3 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧))
76sps 2219 . 2 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧))
84, 7pm2.61d2 174 1 (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wal 1651  wnf 1879  [wsb 2064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-12 2213  ax-13 2377
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065
This theorem is referenced by: (None)
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