Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axsep2 Structured version   Unicode version

Theorem axsep2 4324
 Description: A less restrictive version of the Separation Scheme axsep 4322, 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 4323 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 2497 . . . . . . 7
21anbi1d 686 . . . . . 6
3 anabs5 785 . . . . . 6
42, 3syl6bb 253 . . . . 5
54bibi2d 310 . . . 4
65albidv 1635 . . 3
76exbidv 1636 . 2
8 ax-sep 4323 . 2
97, 8chvarv 1969 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359  wal 1549  wex 1550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323 This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-cleq 2429  df-clel 2432
 Copyright terms: Public domain W3C validator