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Theorem axreg2 7563
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 7562 . 2  |-  ( E. x  x  e.  y  ->  E. x ( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y ) ) )
2119.23bi 1776 1  |-  ( x  e.  y  ->  E. x
( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   A.wal 1550   E.wex 1551
This theorem is referenced by:  zfregcl  7564  axregndlem2  8480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-reg 7562
This theorem depends on definitions:  df-bi 179  df-ex 1552
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