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Theorem axsep2 4079
 Description: A less restrictive version of the Separation Scheme ax-sep 4078, where variables and can both appear free in the wff , which can therefore be thought of as . This version was derived from the more restrictive ax-sep 4078 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
axsep2
Distinct variable groups:   ,,   ,
Allowed substitution hints:   (,)

Proof of Theorem axsep2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eleq2 2218 . . . . . . 7
21anbi1d 461 . . . . . 6
3 anabs5 563 . . . . . 6
42, 3bitrdi 195 . . . . 5
54bibi2d 231 . . . 4
65albidv 1801 . . 3
76exbidv 1802 . 2
8 ax-sep 4078 . 2
97, 8chvarv 1914 1
 Colors of variables: wff set class Syntax hints:   wa 103   wb 104  wal 1330  wex 1469 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-ext 2136  ax-sep 4078 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2147  df-clel 2150 This theorem is referenced by: (None)
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