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Theorem bnd 4169
Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 4117). 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 4115. (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
StepHypRef Expression
1 nfv 1528 . 2  |-  F/ w ph
21ax-coll 4115 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 1492   A.wral 2455   E.wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1449  ax-17 1526  ax-coll 4115
This theorem depends on definitions:  df-bi 117  df-nf 1461
This theorem is referenced by:  bnd2  4170
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