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Theorem ax7 1941
Description: Proof of ax-7 1933 from ax7v1 1935 and ax7v2 1936, proving sufficiency of the conjunction of the latter two weakened versions of ax7v 1934, which is itself a weakened version of ax-7 1933.

Note that the weakened version of ax-7 1933 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 1936 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
2 ax7v2 1936 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑧𝑡 = 𝑧))
3 ax7v1 1935 . . . . . 6 (𝑡 = 𝑦 → (𝑡 = 𝑧𝑦 = 𝑧))
43imp 445 . . . . 5 ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧)
54a1i 11 . . . 4 (𝑥 = 𝑡 → ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧))
61, 2, 5syl2and 500 . . 3 (𝑥 = 𝑡 → ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧))
76expd 452 . 2 (𝑥 = 𝑡 → (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧)))
8 ax6evr 1940 . 2 𝑡 𝑥 = 𝑡
97, 8exlimiiv 1857 1 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703
This theorem is referenced by:  equcomi  1942  equtr  1946  equequ1  1950  equvinv  1957  cbvaev  1977  aeveq  1980  aevOLD  2160  aevALTOLD  2319  axc16i  2320  equvel  2345  axext3  2602  dtru  4848  axextnd  9398  bj-dtru  32772  bj-mo3OLD  32807  wl-aetr  33288  wl-exeq  33292  wl-aleq  33293  wl-nfeqfb  33294  equcomi1  34004  hbequid  34013  equidqe  34026  aev-o  34035  ax6e2eq  38593  ax6e2eqVD  38963
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