MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax13ALT Structured version   Visualization version   GIF version

Theorem ax13ALT 2439
Description: Alternate proof of ax13 2385 from FOL, sp 2181, and axc9 2392. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax13ALT 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Proof of Theorem ax13ALT
StepHypRef Expression
1 sp 2181 . . . 4 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
21con3i 157 . . 3 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3 sp 2181 . . . 4 (∀𝑥 𝑥 = 𝑧𝑥 = 𝑧)
43con3i 157 . . 3 𝑥 = 𝑧 → ¬ ∀𝑥 𝑥 = 𝑧)
5 axc9 2392 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
62, 4, 5syl2im 40 . 2 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
7 ax13b 2039 . 2 ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))))
86, 7mpbir 234 1 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator