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Theorem axpowndlem1 8989
Description: Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
Assertion
Ref Expression
axpowndlem1  |-  ( A. x  x  =  y  ->  ( -.  x  =  y  ->  E. x A. y ( A. x
( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) ) )

Proof of Theorem axpowndlem1
StepHypRef Expression
1 pm2.24 109 . 2  |-  ( x  =  y  ->  ( -.  x  =  y  ->  E. x A. y
( A. x ( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) ) )
21sps 1866 1  |-  ( A. x  x  =  y  ->  ( -.  x  =  y  ->  E. x A. y ( A. x
( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1393   E.wex 1613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-12 1855
This theorem depends on definitions:  df-bi 185  df-ex 1614
This theorem is referenced by:  axpowndlem3OLD  8993  axpownd  8995
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