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Theorem axc9 2194
Description: Derive set.mm's original ax-c9 33068 from the shorter ax-13 2137. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.)
Assertion
Ref Expression
axc9 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Proof of Theorem axc9
StepHypRef Expression
1 nfeqf 2193 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)
21nfrd 2006 . 2 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
32ex 448 1 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-nf 1699
This theorem is referenced by:  ax13OLD  2197  hbae  2207  axi12  2492  axbnd  2493  axext4dist  30793  bj-ax6elem1  31675  axc11n11r  31695  bj-hbaeb2  31835  wl-aleq  32375  ax12eq  33119  ax12indalem  33123
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