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Theorem ax7 1930
 Description: Proof of ax-7 1922 from ax7v1 1924 and ax7v2 1925, proving sufficiency of the conjunction of the latter two weakened versions of ax7v 1923, which is itself a weakened version of ax-7 1922. Note that the weakened version of ax-7 1922 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 1925 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
2 ax7v2 1925 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑧𝑡 = 𝑧))
3 ax7v1 1924 . . . . . 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 1929 . 2 𝑡 𝑥 = 𝑡
97, 8exlimiiv 1846 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 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by:  equcomi  1931  equtr  1935  equequ1  1939  equvinv  1946  cbvaev  1966  aeveq  1969  aevOLD  2148  aevALTOLD  2309  axc16i  2310  equvel  2335  axext3  2592  dtru  4783  axextnd  9292  2spotmdisj  26595  bj-dtru  31985  bj-mo3OLD  32022  wl-aetr  32496  wl-exeq  32500  wl-aleq  32501  wl-nfeqfb  32502  equcomi1  33203  hbequid  33212  equidqe  33225  aev-o  33234  ax6e2eq  37794  ax6e2eqVD  38165
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