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Theorem ax7 1989
Description: Proof of ax-7 1981 from ax7v1 1983 and ax7v2 1984, proving sufficiency of the conjunction of the latter two weakened versions of ax7v 1982, which is itself a weakened version of ax-7 1981.

Note that the weakened version of ax-7 1981 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 1984 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
2 ax7v2 1984 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑧𝑡 = 𝑧))
3 ax7v1 1983 . . . . . 6 (𝑡 = 𝑦 → (𝑡 = 𝑧𝑦 = 𝑧))
43imp 444 . . . . 5 ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧)
54a1i 11 . . . 4 (𝑥 = 𝑡 → ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧))
61, 2, 5syl2and 499 . . 3 (𝑥 = 𝑡 → ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧))
76expd 451 . 2 (𝑥 = 𝑡 → (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧)))
8 ax6evr 1988 . 2 𝑡 𝑥 = 𝑡
97, 8exlimiiv 1899 1 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  equcomi  1990  equtr  1994  equequ1  1998  equvinv  2003  cbvaev  2021  aeveq  2024  axc16i  2353  equvel  2375  axext3  2633  dtru  4887  axextnd  9451  bj-dtru  32922  bj-mo3OLD  32957  wl-aetr  33447  wl-exeq  33451  wl-aleq  33452  wl-nfeqfb  33453  equcomi1  34504  hbequid  34513  equidqe  34526  aev-o  34535  ax6e2eq  39090  ax6e2eqVD  39457
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