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Theorem ax8 1983
 Description: Proof of ax-8 1979 from ax8v1 1981 and ax8v2 1982, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 1980, which is itself a weakened version of ax-8 1979. (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 1946 . 2 (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥𝑡 = 𝑦))
2 ax8v2 1982 . . . . 5 (𝑥 = 𝑡 → (𝑥𝑧𝑡𝑧))
32equcoms 1934 . . . 4 (𝑡 = 𝑥 → (𝑥𝑧𝑡𝑧))
4 ax8v1 1981 . . . 4 (𝑡 = 𝑦 → (𝑡𝑧𝑦𝑧))
53, 4sylan9 687 . . 3 ((𝑡 = 𝑥𝑡 = 𝑦) → (𝑥𝑧𝑦𝑧))
65exlimiv 1845 . 2 (∃𝑡(𝑡 = 𝑥𝑡 = 𝑦) → (𝑥𝑧𝑦𝑧))
71, 6sylbi 206 1 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by:  elequ1  1984  el  4773  axextdfeq  30947  ax8dfeq  30948  exnel  30952  bj-ax89  31854  bj-el  31984
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