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Theorem ax8 2163
Description: Proof of ax-8 2159 from ax8v1 2161 and ax8v2 2162, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 2160, which is itself a weakened version of ax-8 2159. (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 2129 . 2 (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥𝑡 = 𝑦))
2 ax8v2 2162 . . . . 5 (𝑥 = 𝑡 → (𝑥𝑧𝑡𝑧))
32equcoms 2119 . . . 4 (𝑡 = 𝑥 → (𝑥𝑧𝑡𝑧))
4 ax8v1 2161 . . . 4 (𝑡 = 𝑦 → (𝑡𝑧𝑦𝑧))
53, 4sylan9 504 . . 3 ((𝑡 = 𝑥𝑡 = 𝑦) → (𝑥𝑧𝑦𝑧))
65exlimiv 2026 . 2 (∃𝑡(𝑡 = 𝑥𝑡 = 𝑦) → (𝑥𝑧𝑦𝑧))
71, 6sylbi 209 1 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wex 1875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876
This theorem is referenced by:  elequ1  2164  el  5038  axextdfeq  32208  ax8dfeq  32209  exnel  32213  bj-ax89  33166  bj-el  33285
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