Theorem axreg2 8498

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

Step	Hyp	Ref	Expression
1		ax-reg 8497	. 2 |- ( E. x x e. y -> E. x ( x e. y /\ A. z ( z e. x -> -. z e. y ) ) )
2	1	19.23bi 2061	1 |- ( x e. y -> E. x ( x e. y /\ A. z ( z e. x -> -. z e. y ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  zfregcl  8499  zfregclOLD  8501  axregndlem2  9425