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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 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 2013 . . 3 𝑡 𝑥 = 𝑡
86, 7exlimiiv 1923 . 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 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772
This theorem is referenced by:  equcomi  2015  equtr  2019  equequ1  2023  cbvaev  2049  aeveq  2052  axc16i  2450  equvel  2471  mo4  2643  axextnd  10001  bj-dtru  34036  wl-aetr  34651  wl-exeq  34655  wl-aleq  34656  wl-nfeqfb  34657  equcomi1  35916  hbequid  35925  equidqe  35938  aev-o  35947  ax6e2eq  40768  ax6e2eqVD  41118  2reu8i  43189
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