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Theorem bnd 7564
Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 5750), derived from the Collection Principle cp 7563. 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 7563 . . 3  |-  E. w A. x  e.  z 
( E. y ph  ->  E. y  e.  w  ph )
2 ralim 2616 . . . 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 1565 . . 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 1843 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 1530   A.wral 2545   E.wrex 2546
This theorem is referenced by:  bnd2  7565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-reg 7308  ax-inf2 7344
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-recs 6390  df-rdg 6425  df-r1 7438  df-rank 7439
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