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Theorem ax9 2043
 Description: Proof of ax-9 2039 from ax9v1 2041 and ax9v2 2042, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2040, which is itself a weakened version of ax-9 2039. (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 2003 . 2 (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥𝑡 = 𝑦))
2 ax9v2 2042 . . . . 5 (𝑥 = 𝑡 → (𝑧𝑥𝑧𝑡))
32equcoms 1993 . . . 4 (𝑡 = 𝑥 → (𝑧𝑥𝑧𝑡))
4 ax9v1 2041 . . . 4 (𝑡 = 𝑦 → (𝑧𝑡𝑧𝑦))
53, 4sylan9 690 . . 3 ((𝑡 = 𝑥𝑡 = 𝑦) → (𝑧𝑥𝑧𝑦))
65exlimiv 1898 . 2 (∃𝑡(𝑡 = 𝑥𝑡 = 𝑦) → (𝑧𝑥𝑧𝑦))
71, 6sylbi 207 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 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-9 2039 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745 This theorem is referenced by:  elequ2  2044  el  4877  dtru  4887  fv3  6244  elirrv  8542  bj-ax89  32792  bj-el  32921  bj-dtru  32922  axc11next  38924
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