Theorem zfrep6 4099

Description: A version of the Axiom of Replacement. Normally ph would have free variables x and y. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4100 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version. (Contributed by NM, 10-Oct-2003.)

Assertion

Ref	Expression
zfrep6	|- ( A. x e. z E! y ph -> E. w A. x e. z E. y e. w ph )

Distinct variable groups:   ph, w   x, y, z, w

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem zfrep6

Step	Hyp	Ref	Expression
1		nfv 1516	. 2 |- F/ w ph
2	1	repizf 4098	1 |- ( A. x e. z E! y ph -> E. w A. x e. z E. y e. w ph )

Syntax hints:   -> wi 4   E.wex 1480   E!weu 2014   A.wral 2444   E.wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-coll 4097

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-ral 2449

This theorem is referenced by:  funimaexglem  5271