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Theorem ax7 2023
 Description: Proof of ax-7 2015 from ax7v1 2017 and ax7v2 2018 (and earlier axioms), proving sufficiency of the conjunction of the latter two weakened versions of ax7v 2016, which is itself a weakened version of ax-7 2015. Note that the weakened version of ax-7 2015 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 2018 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
2 ax7v2 2018 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑧𝑡 = 𝑧))
3 ax7v1 2017 . . . . . 6 (𝑡 = 𝑦 → (𝑡 = 𝑧𝑦 = 𝑧))
43imp 410 . . . . 5 ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧)
54a1i 11 . . . 4 (𝑥 = 𝑡 → ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧))
61, 2, 5syl2and 610 . . 3 (𝑥 = 𝑡 → ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧))
7 ax6evr 2022 . . 3 𝑡 𝑥 = 𝑡
86, 7exlimiiv 1932 . 2 ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
98ex 416 1 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  equcomi  2024  equtr  2028  equequ1  2032  cbvaev  2058  aeveq  2061  axc16i  2447  equvel  2468  mo4  2625  axextnd  10004  bj-dtru  34270  wl-aetr  34950  wl-exeq  34955  wl-aleq  34956  wl-nfeqfb  34957  equcomi1  36212  hbequid  36221  equidqe  36234  aev-o  36243  ax6e2eq  41278  ax6e2eqVD  41628  2reu8i  43684
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