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Theorem bnd 7088
Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 5241), derived from the Collection Principle cp 7087. 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. (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 cp 7087 . . 3  |-  E. w A. x  e.  z 
( E. y ph  ->  E. y  e.  w  ph )
2 ralim 2373 . . . 4  |-  ( A. x  e.  z  ( E. y ph  ->  E. y  e.  w  ph )  -> 
( A. x  e.  z  E. y ph  ->  A. x  e.  z  E. y  e.  w  ph ) )
32eximi 1484 . . 3  |-  ( E. w A. x  e.  z  ( E. y ph  ->  E. y  e.  w  ph )  ->  E. w
( A. x  e.  z  E. y ph  ->  A. x  e.  z  E. y  e.  w  ph ) )
41, 3ax-mp 8 . 2  |-  E. w
( A. x  e.  z  E. y ph  ->  A. x  e.  z  E. y  e.  w  ph )
5419.37aiv 1889 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 1456   A.wral 2311   E.wrex 2312
This theorem is referenced by:  bnd2  7089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1452  ax-6 1453  ax-7 1454  ax-gen 1455  ax-8 1535  ax-11 1536  ax-13 1537  ax-14 1538  ax-17 1540  ax-12o 1574  ax-10 1588  ax-9 1594  ax-4 1601  ax-16 1787  ax-ext 2082  ax-rep 3735  ax-sep 3745  ax-nul 3753  ax-pow 3789  ax-pr 3813  ax-un 4105  ax-reg 6833  ax-inf2 6869
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 901  df-3an 902  df-ex 1457  df-sb 1748  df-eu 1970  df-mo 1971  df-clab 2088  df-cleq 2093  df-clel 2096  df-ne 2220  df-ral 2315  df-rex 2316  df-reu 2317  df-rab 2318  df-v 2514  df-sbc 2688  df-csb 2770  df-dif 2833  df-un 2835  df-in 2837  df-ss 2841  df-pss 2843  df-nul 3111  df-if 3221  df-pw 3282  df-sn 3300  df-pr 3301  df-tp 3302  df-op 3303  df-uni 3469  df-int 3503  df-iun 3546  df-iin 3547  df-br 3631  df-opab 3685  df-mpt 3686  df-tr 3718  df-eprel 3900  df-id 3904  df-po 3909  df-so 3910  df-fr 3947  df-we 3949  df-ord 3990  df-on 3991  df-lim 3992  df-suc 3993  df-om 4268  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-fun 4322  df-fn 4323  df-f 4324  df-f1 4325  df-fo 4326  df-f1o 4327  df-fv 4328  df-recs 5887  df-rdg 5922  df-r1 6963  df-rank 6964
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