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Theorem ax8 1993
Description: Proof of ax-8 1989 from ax8v1 1991 and ax8v2 1992, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 1990, which is itself a weakened version of ax-8 1989. (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 1956 . 2 (𝑥 = 𝑦 ↔ ∃𝑡(𝑡 = 𝑥𝑡 = 𝑦))
2 ax8v2 1992 . . . . 5 (𝑥 = 𝑡 → (𝑥𝑧𝑡𝑧))
32equcoms 1944 . . . 4 (𝑡 = 𝑥 → (𝑥𝑧𝑡𝑧))
4 ax8v1 1991 . . . 4 (𝑡 = 𝑦 → (𝑡𝑧𝑦𝑧))
53, 4sylan9 688 . . 3 ((𝑡 = 𝑥𝑡 = 𝑦) → (𝑥𝑧𝑦𝑧))
65exlimiv 1855 . 2 (∃𝑡(𝑡 = 𝑥𝑡 = 𝑦) → (𝑥𝑧𝑦𝑧))
71, 6sylbi 207 1 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by:  elequ1  1994  el  4817  axextdfeq  31457  ax8dfeq  31458  exnel  31462  bj-ax89  32362  bj-el  32492
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