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Theorem ax7 2115
 Description: Proof of ax-7 2107 from ax7v1 2109 and ax7v2 2110 (and earlier axioms), proving sufficiency of the conjunction of the latter two weakened versions of ax7v 2108, which is itself a weakened version of ax-7 2107. Note that the weakened version of ax-7 2107 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 2110 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑦𝑡 = 𝑦))
2 ax7v2 2110 . . . 4 (𝑥 = 𝑡 → (𝑥 = 𝑧𝑡 = 𝑧))
3 ax7v1 2109 . . . . . 6 (𝑡 = 𝑦 → (𝑡 = 𝑧𝑦 = 𝑧))
43imp 396 . . . . 5 ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧)
54a1i 11 . . . 4 (𝑥 = 𝑡 → ((𝑡 = 𝑦𝑡 = 𝑧) → 𝑦 = 𝑧))
61, 2, 5syl2and 602 . . 3 (𝑥 = 𝑡 → ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧))
7 ax6evr 2114 . . 3 𝑡 𝑥 = 𝑡
86, 7exlimiiv 2027 . 2 ((𝑥 = 𝑦𝑥 = 𝑧) → 𝑦 = 𝑧)
98ex 402 1 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107 This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876 This theorem is referenced by:  equcomi  2116  equtr  2120  equequ1  2124  equvinvOLD  2130  cbvaev  2148  aeveq  2151  axc16i  2444  equvel  2464  axext3  2780  dtru  5041  axextnd  9702  bj-dtru  33292  bj-mo3OLD  33326  wl-aetr  33806  wl-exeq  33810  wl-aleq  33811  wl-nfeqfb  33812  equcomi1  34920  hbequid  34929  equidqe  34942  aev-o  34951  ax6e2eq  39538  ax6e2eqVD  39898
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