Theorem bnd 4054

Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 4003). Its strength lies in the rather profound fact that ph ( x , y ) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. In the context of IZF, it is just a slight variation of ax-coll 4001. (Contributed by NM, 17-Oct-2004.)

Assertion

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

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

Allowed substitution hints:   ph( x, y)

Proof of Theorem bnd

Step	Hyp	Ref	Expression
1		nfv 1489	. 2 |- F/ w ph
2	1	ax-coll 4001	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 1449   A.wral 2388   E.wrex 2389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1406  ax-17 1487  ax-coll 4001

This theorem depends on definitions:  df-bi 116  df-nf 1418

This theorem is referenced by:  bnd2  4055