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

Step	Hyp	Ref	Expression
1		nfeqf 2193	. . 3 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)
2	1	nfrd 2006	. 2 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
3	2	ex 448	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

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