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Theorem axreg2 8008
Description: Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
axreg2  |-  ( x  e.  y  ->  E. x
( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y
) ) )
Distinct variable group:    x, y, z

Proof of Theorem axreg2
StepHypRef Expression
1 ax-reg 8007 . 2  |-  ( E. x  x  e.  y  ->  E. x ( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y ) ) )
2119.23bi 1814 1  |-  ( x  e.  y  ->  E. x
( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1372   E.wex 1591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-12 1798  ax-reg 8007
This theorem depends on definitions:  df-bi 185  df-ex 1592
This theorem is referenced by:  zfregcl  8009  axregndlem2  8969
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