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Theorem a9evsep 3907
 Description: Derive a weakened version of ax-i9 1439, where and must be distinct, from Separation ax-sep 3903 and Extensionality ax-ext 2038. The theorem also holds (ax9vsep 3908), 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 3903 . 2
2 id 19 . . . . . . . 8
32biantru 290 . . . . . . 7
43bibi2i 220 . . . . . 6
54biimpri 128 . . . . 5
65alimi 1360 . . . 4
7 ax-ext 2038 . . . 4
86, 7syl 14 . . 3
98eximi 1507 . 2
101, 9ax-mp 7 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101   wb 102  wal 1257   wceq 1259  wex 1397   wcel 1409 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443  ax-ext 2038  ax-sep 3903 This theorem depends on definitions:  df-bi 114 This theorem is referenced by:  ax9vsep  3908
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