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Theorem ax9 1990
Description: Proof of ax-9 1986 from ax9v1 1988 and ax9v2 1989, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 1987, which is itself a weakened version of ax-9 1986. (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 1946 . 2 (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥𝑡 = 𝑦))
2 ax9v2 1989 . . . . 5 (𝑥 = 𝑡 → (𝑧𝑥𝑧𝑡))
32equcoms 1934 . . . 4 (𝑡 = 𝑥 → (𝑧𝑥𝑧𝑡))
4 ax9v1 1988 . . . 4 (𝑡 = 𝑦 → (𝑧𝑡𝑧𝑦))
53, 4sylan9 687 . . 3 ((𝑡 = 𝑥𝑡 = 𝑦) → (𝑧𝑥𝑧𝑦))
65exlimiv 1845 . 2 (∃𝑡(𝑡 = 𝑥𝑡 = 𝑦) → (𝑧𝑥𝑧𝑦))
71, 6sylbi 206 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
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-9 1986
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696
This theorem is referenced by:  elequ2  1991  el  4773  dtru  4783  fv3  6116  elirrv  8387  bj-ax89  31854  bj-el  31984  bj-dtru  31985  axc11next  37629
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