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Theorem bnd 8104
Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 6550), derived from the Collection Principle cp 8103. 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 8103 . . 3  |-  E. w A. x  e.  z 
( E. y ph  ->  E. y  e.  w  ph )
2 ralim 2792 . . 3  |-  ( 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 ) )
31, 2eximii 1627 . 2  |-  E. w
( A. x  e.  z  E. y ph  ->  A. x  e.  z  E. y  e.  w  ph )
4319.37aiv 1919 1  |-  ( A. x  e.  z  E. y ph  ->  E. w A. x  e.  z  E. y  e.  w  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wex 1586   A.wral 2720   E.wrex 2721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-reg 7812  ax-inf2 7852
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-om 6482  df-recs 6837  df-rdg 6871  df-r1 7976  df-rank 7977
This theorem is referenced by:  bnd2  8105
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