Theorem axpowndlem1 3743

Description: Lemma for the Axiom of Power Sets with no distinct variable conditions.

Assertion

Ref	Expression
axpowndlem1	|- (A.x x = y -> (-. x = y -> E.xA.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 72	. 2 |- (x = y -> (-. x = y -> E.xA.y(A.x(E.z x e. y -> A.y x e. z) -> y e. x)))
2	1	a4s 682	1 |- (A.x x = y -> (-. x = y -> E.xA.y(A.x(E.z x e. y -> A.y x e. z) -> y e. x)))

Syntax hints:  -. wn 1   -> wi 2  A.wal 672  E.wex 678   = weq 797   e. wel 803

This theorem is referenced by:  axpowndlem3 3745  axpownd 3747

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673

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