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Theorem ax7 1929
Description: Proof of ax-7 1921 from ax7v1 1923 and ax7v2 1924, proving sufficiency of the conjunction of the latter two weakened versions of ax7v 1922, which is itself a weakened version of ax-7 1921.

Note that the weakened version of ax-7 1921 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 1924 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
2 ax7v2 1924 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑧𝑡 = 𝑧))
3 ax7v1 1923 . . . . . 6 (𝑡 = 𝑦 → (𝑡 = 𝑧𝑦 = 𝑧))
43imp 443 . . . . 5 ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧)
54a1i 11 . . . 4 (𝑥 = 𝑡 → ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧))
61, 2, 5syl2and 498 . . 3 (𝑥 = 𝑡 → ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧))
76expd 450 . 2 (𝑥 = 𝑡 → (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧)))
8 ax6evr 1928 . 2 𝑡 𝑥 = 𝑡
97, 8exlimiiv 1845 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 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  equcomi  1930  equtr  1934  equequ1  1938  equvinv  1945  cbvaev  1965  aeveq  1968  aevOLD  2147  aevALTOLD  2308  axc16i  2309  equvel  2334  axext3  2591  dtru  4778  axextnd  9269  2spotmdisj  26361  bj-dtru  31791  bj-mo3OLD  31828  wl-aetr  32292  wl-exeq  32296  wl-aleq  32297  wl-nfeqfb  32298  equcomi1  32999  hbequid  33008  equidqe  33021  aev-o  33030  ax6e2eq  37590  ax6e2eqVD  37961
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