 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax9 Structured version   Visualization version   GIF version

Theorem ax9 1989
 Description: Proof of ax-9 1985 from ax9v1 1987 and ax9v2 1988, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 1986, which is itself a weakened version of ax-9 1985. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)
Assertion
Ref Expression
ax9 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Proof of Theorem ax9
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 equvinv 1945 . 2 (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥𝑡 = 𝑦))
2 ax9v2 1988 . . . . 5 (𝑥 = 𝑡 → (𝑧𝑥𝑧𝑡))
32equcoms 1933 . . . 4 (𝑡 = 𝑥 → (𝑧𝑥𝑧𝑡))
4 ax9v1 1987 . . . 4 (𝑡 = 𝑦 → (𝑧𝑡𝑧𝑦))
53, 4sylan9 686 . . 3 ((𝑡 = 𝑥𝑡 = 𝑦) → (𝑧𝑥𝑧𝑦))
65exlimiv 1844 . 2 (∃𝑡(𝑡 = 𝑥𝑡 = 𝑦) → (𝑧𝑥𝑧𝑦))
71, 6sylbi 205 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382  ∃wex 1694 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985 This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695 This theorem is referenced by:  elequ2  1990  el  4768  dtru  4778  fv3  6101  elirrv  8364  bj-ax89  31660  bj-el  31790  bj-dtru  31791  axc11next  37425
 Copyright terms: Public domain W3C validator