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Theorem ax8 2113
Description: Proof of ax-8 2109 from ax8v1 2111 and ax8v2 2112, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 2110, which is itself a weakened version of ax-8 2109. (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 2029 . 2 (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥𝑡 = 𝑦))
2 ax8v2 2112 . . . . 5 (𝑥 = 𝑡 → (𝑥𝑧𝑡𝑧))
32equcoms 2020 . . . 4 (𝑡 = 𝑥 → (𝑥𝑧𝑡𝑧))
4 ax8v1 2111 . . . 4 (𝑡 = 𝑦 → (𝑡𝑧𝑦𝑧))
53, 4sylan9 508 . . 3 ((𝑡 = 𝑥𝑡 = 𝑦) → (𝑥𝑧𝑦𝑧))
65exlimiv 1924 . 2 (∃𝑡(𝑡 = 𝑥𝑡 = 𝑦) → (𝑥𝑧𝑦𝑧))
71, 6sylbi 218 1 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774
This theorem is referenced by:  elequ1  2114  ax9ALT  2822  el  5267  axextdfeq  32945  ax8dfeq  32946  exnel  32950  bj-ax89  33914
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