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Theorem ax9 2113
Description: Proof of ax-9 2109 from ax9v1 2111 and ax9v2 2112, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2110, which is itself a weakened version of ax-9 2109. (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 2025 . 2 (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥𝑡 = 𝑦))
2 ax9v2 2112 . . . . 5 (𝑥 = 𝑡 → (𝑧𝑥𝑧𝑡))
32equcoms 2016 . . . 4 (𝑡 = 𝑥 → (𝑧𝑥𝑧𝑡))
4 ax9v1 2111 . . . 4 (𝑡 = 𝑦 → (𝑧𝑡𝑧𝑦))
53, 4sylan9 506 . . 3 ((𝑡 = 𝑥𝑡 = 𝑦) → (𝑧𝑥𝑧𝑦))
65exlimiv 1926 . 2 (∃𝑡(𝑡 = 𝑥𝑡 = 𝑦) → (𝑧𝑥𝑧𝑦))
71, 6sylbi 216 1 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wex 1774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775
This theorem is referenced by:  elequ2  2114  elALT2  5373  fv3  6919  elirrv  9639  bj-ax89  36382  axc11next  44080
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