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Theorem ax7 2014
Description: Proof of ax-7 2006 from ax7v1 2008 and ax7v2 2009 (and earlier axioms), proving sufficiency of the conjunction of the latter two weakened versions of ax7v 2007, which is itself a weakened version of ax-7 2006.

Note that the weakened version of ax-7 2006 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 2009 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
2 ax7v2 2009 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑧𝑡 = 𝑧))
3 ax7v1 2008 . . . . . 6 (𝑡 = 𝑦 → (𝑡 = 𝑧𝑦 = 𝑧))
43imp 406 . . . . 5 ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧)
54a1i 11 . . . 4 (𝑥 = 𝑡 → ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧))
61, 2, 5syl2and 608 . . 3 (𝑥 = 𝑡 → ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧))
7 ax6evr 2013 . . 3 𝑡 𝑥 = 𝑡
86, 7exlimiiv 1930 . 2 ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
98ex 412 1 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779
This theorem is referenced by:  equcomi  2015  equtr  2019  equequ1  2023  cbvaev  2052  aeveq  2055  axc16i  2440  equvel  2460  mo4  2565  axextnd  10632  in-ax8  36226  ss-ax8  36227  wl-aetr  37531  wl-exeq  37536  wl-aleq  37537  wl-nfeqfb  37538  equcomi1  38902  hbequid  38911  equidqe  38924  aev-o  38933  ax6e2eq  44582  ax6e2eqVD  44932  et-equeucl  46892  2reu8i  47130
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