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Theorem ax9 2095
 Description: Proof of ax-9 2091 from ax9v1 2093 and ax9v2 2094, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2092, which is itself a weakened version of ax-9 2091. (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 2013 . 2 (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥𝑡 = 𝑦))
2 ax9v2 2094 . . . . 5 (𝑥 = 𝑡 → (𝑧𝑥𝑧𝑡))
32equcoms 2004 . . . 4 (𝑡 = 𝑥 → (𝑧𝑥𝑧𝑡))
4 ax9v1 2093 . . . 4 (𝑡 = 𝑦 → (𝑧𝑡𝑧𝑦))
53, 4sylan9 508 . . 3 ((𝑡 = 𝑥𝑡 = 𝑦) → (𝑧𝑥𝑧𝑦))
65exlimiv 1908 . 2 (∃𝑡(𝑡 = 𝑥𝑡 = 𝑦) → (𝑧𝑥𝑧𝑦))
71, 6sylbi 218 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1761 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-9 2091 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762 This theorem is referenced by:  elequ2  2096  el  5161  fv3  6556  elirrv  8906  bj-ax89  33610  bj-el  33703  bj-dtru  33704  axc11next  40276
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