MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc9 Structured version   Visualization version   GIF version

Theorem axc9 2382
Description: Derive set.mm's original ax-c9 36527 from the shorter ax-13 2372. Usage is discouraged to avoid uninformed ax-13 2372 propagation. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.) (New usage is discouraged.)
Assertion
Ref Expression
axc9 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Proof of Theorem axc9
StepHypRef Expression
1 nfeqf 2381 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)
21nf5rd 2198 . 2 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
32ex 416 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 399  wal 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-12 2179  ax-13 2372
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791
This theorem is referenced by:  ax13ALT  2425  hbae  2431  axi12  2708  axextbdist  33348  bj-ax6elem1  34485  axc11n11r  34503  bj-hbaeb2  34632  wl-aleq  35317  ax12eq  36578  ax12indalem  36582
  Copyright terms: Public domain W3C validator