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Theorem ax7 2020
Description: Proof of ax-7 2012 from ax7v1 2014 and ax7v2 2015 (and earlier axioms), proving sufficiency of the conjunction of the latter two weakened versions of ax7v 2013, which is itself a weakened version of ax-7 2012.

Note that the weakened version of ax-7 2012 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 2015 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
2 ax7v2 2015 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑧𝑡 = 𝑧))
3 ax7v1 2014 . . . . . 6 (𝑡 = 𝑦 → (𝑡 = 𝑧𝑦 = 𝑧))
43imp 408 . . . . 5 ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧)
54a1i 11 . . . 4 (𝑥 = 𝑡 → ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧))
61, 2, 5syl2and 609 . . 3 (𝑥 = 𝑡 → ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧))
7 ax6evr 2019 . . 3 𝑡 𝑥 = 𝑡
86, 7exlimiiv 1935 . 2 ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
98ex 414 1 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783
This theorem is referenced by:  equcomi  2021  equtr  2025  equequ1  2029  cbvaev  2057  aeveq  2060  axc16i  2436  equvel  2456  mo4  2561  axextnd  10586  wl-aetr  36398  wl-exeq  36403  wl-aleq  36404  wl-nfeqfb  36405  equcomi1  37770  hbequid  37779  equidqe  37792  aev-o  37801  ax6e2eq  43318  ax6e2eqVD  43668  et-equeucl  45588  2reu8i  45821
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