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Theorem ax7 2001
 Description: Proof of ax-7 1993 from ax7v1 1995 and ax7v2 1996 (and earlier axioms), proving sufficiency of the conjunction of the latter two weakened versions of ax7v 1994, which is itself a weakened version of ax-7 1993. Note that the weakened version of ax-7 1993 obtained by adding a disjoint variable 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 1996 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
2 ax7v2 1996 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑧𝑡 = 𝑧))
3 ax7v1 1995 . . . . . 6 (𝑡 = 𝑦 → (𝑡 = 𝑧𝑦 = 𝑧))
43imp 407 . . . . 5 ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧)
54a1i 11 . . . 4 (𝑥 = 𝑡 → ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧))
61, 2, 5syl2and 607 . . 3 (𝑥 = 𝑡 → ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧))
7 ax6evr 2000 . . 3 𝑡 𝑥 = 𝑡
86, 7exlimiiv 1910 . 2 ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
98ex 413 1 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1763 This theorem is referenced by:  equcomi  2002  equtr  2006  equequ1  2010  cbvaev  2030  aeveq  2033  axc16i  2414  equvel  2435  mo4  2606  axextnd  9862  bj-dtru  33702  wl-aetr  34315  wl-exeq  34319  wl-aleq  34320  wl-nfeqfb  34321  equcomi1  35580  hbequid  35589  equidqe  35602  aev-o  35611  ax6e2eq  40443  ax6e2eqVD  40793  2reu8i  42842
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