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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.)
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
StepHypRef Expression
1 nfv 1489 . 2  |-  F/ w ph
21ax-coll 4001 1  |-  ( A. x  e.  z  E. y ph  ->  E. w A. x  e.  z  E. y  e.  w  ph )
Colors of variables: wff set class
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
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