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

Proof of Theorem a9evsep
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax-sep 3866 . 2
2 id 19 . . . . . . . 8
32biantru 286 . . . . . . 7
43bibi2i 216 . . . . . 6
54biimpri 124 . . . . 5
65alimi 1341 . . . 4
7 ax-ext 2019 . . . 4
86, 7syl 14 . . 3
98eximi 1488 . 2
101, 9ax-mp 7 1
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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