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Theorem a9evsep 3870
 Description: Derive a weakened version of ax-i9 1420, where x and y must be distinct, from Separation ax-sep 3866 and Extensionality ax-ext 2019. The theorem ¬ ∀x¬ x = y also holds (ax9vsep 3871), but in intuitionistic logic ∃xx = y is stronger. (Contributed by Jim Kingdon, 25-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a9evsep x x = y
Distinct variable group:   x,y

Proof of Theorem a9evsep
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 ax-sep 3866 . 2 xz(z x ↔ (z y (z = zz = z)))
2 id 19 . . . . . . . 8 (z = zz = z)
32biantru 286 . . . . . . 7 (z y ↔ (z y (z = zz = z)))
43bibi2i 216 . . . . . 6 ((z xz y) ↔ (z x ↔ (z y (z = zz = z))))
54biimpri 124 . . . . 5 ((z x ↔ (z y (z = zz = z))) → (z xz y))
65alimi 1341 . . . 4 (z(z x ↔ (z y (z = zz = z))) → z(z xz y))
7 ax-ext 2019 . . . 4 (z(z xz y) → x = y)
86, 7syl 14 . . 3 (z(z x ↔ (z y (z = zz = z))) → x = y)
98eximi 1488 . 2 (xz(z x ↔ (z y (z = zz = z))) → x x = y)
101, 9ax-mp 7 1 x x = y
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424  ax-ext 2019  ax-sep 3866 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  ax9vsep  3871
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