Theorem axpowndlem1 9419

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

Step	Hyp	Ref	Expression
1		pm2.24 121	. 2 |- ( x = y -> ( -. x = y -> E. x A. y ( A. x ( E. z x e. y -> A. y x e. z ) -> y e. x ) ) )
2	1	sps 2055	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 ) ) )

Syntax hints:   -. wn 3   -> wi 4   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

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

This theorem is referenced by:  axpownd  9423