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Theorem axc9 2439
Description: Derive set.mm's original ax-c9 34671 from the shorter ax-13 2383. (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 2438 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)
21nf5rd 2205 . 2 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
32ex 449 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 383  wal 1622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-10 2160  ax-12 2188  ax-13 2383
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851
This theorem is referenced by:  ax13ALT  2442  hbae  2449  axi12  2730  axbnd  2731  axext4dist  32003  bj-ax6elem1  32949  axc11n11r  32971  bj-hbaeb2  33103  wl-aleq  33627  ax12eq  34722  ax12indalem  34726
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