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Theorem axsep2 4142
 Description: A less restrictive version of the Separation Scheme axsep 4140, 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 4141 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 2344 . . . . . . 7
21anbi1d 685 . . . . . 6
3 anabs5 784 . . . . . 6
42, 3syl6bb 252 . . . . 5
54bibi2d 309 . . . 4
65albidv 1611 . . 3
76exbidv 1612 . 2
8 ax-sep 4141 . 2
97, 8chvarv 1953 1
 Colors of variables: wff set class Syntax hints:   wb 176   wa 358  wal 1527  wex 1528   wceq 1623   wcel 1684 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-cleq 2276  df-clel 2279
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