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Theorem ax7 2100
Description: Proof of ax-7 2092 from ax7v1 2094 and ax7v2 2095, proving sufficiency of the conjunction of the latter two weakened versions of ax7v 2093, which is itself a weakened version of ax-7 2092.

Note that the weakened version of ax-7 2092 obtained by adding a dv condition on 𝑥, 𝑧 (resp. on 𝑦, 𝑧) does not permit, together with the other axioms, to prove reflexivity (resp. symmetry). (Contributed by BJ, 7-Dec-2020.)

Assertion
Ref Expression
ax7 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Proof of Theorem ax7
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 ax7v2 2095 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
2 ax7v2 2095 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑧𝑡 = 𝑧))
3 ax7v1 2094 . . . . . 6 (𝑡 = 𝑦 → (𝑡 = 𝑧𝑦 = 𝑧))
43imp 393 . . . . 5 ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧)
54a1i 11 . . . 4 (𝑥 = 𝑡 → ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧))
61, 2, 5syl2and 587 . . 3 (𝑥 = 𝑡 → ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧))
7 ax6evr 2099 . . 3 𝑡 𝑥 = 𝑡
86, 7exlimiiv 2010 . 2 ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
98ex 397 1 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852
This theorem is referenced by:  equcomi  2101  equtr  2105  equequ1  2109  equvinvOLD  2115  cbvaev  2135  aeveq  2138  axc16i  2471  equvel  2492  axext3  2752  dtru  4985  axextnd  9614  bj-dtru  33127  bj-mo3OLD  33161  wl-aetr  33645  wl-exeq  33649  wl-aleq  33650  wl-nfeqfb  33651  equcomi1  34701  hbequid  34710  equidqe  34723  aev-o  34732  ax6e2eq  39292  ax6e2eqVD  39659
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