MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax8 Structured version   Visualization version   GIF version

Theorem ax8 2036
Description: Proof of ax-8 2032 from ax8v1 2034 and ax8v2 2035, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 2033, which is itself a weakened version of ax-8 2032. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)
Assertion
Ref Expression
ax8 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Proof of Theorem ax8
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 equvinv 2003 . 2 (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥𝑡 = 𝑦))
2 ax8v2 2035 . . . . 5 (𝑥 = 𝑡 → (𝑥𝑧𝑡𝑧))
32equcoms 1993 . . . 4 (𝑡 = 𝑥 → (𝑥𝑧𝑡𝑧))
4 ax8v1 2034 . . . 4 (𝑡 = 𝑦 → (𝑡𝑧𝑦𝑧))
53, 4sylan9 690 . . 3 ((𝑡 = 𝑥𝑡 = 𝑦) → (𝑥𝑧𝑦𝑧))
65exlimiv 1898 . 2 (∃𝑡(𝑡 = 𝑥𝑡 = 𝑦) → (𝑥𝑧𝑦𝑧))
71, 6sylbi 207 1 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  elequ1  2037  el  4877  axextdfeq  31827  ax8dfeq  31828  exnel  31832  bj-ax89  32792  bj-el  32921
  Copyright terms: Public domain W3C validator