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Theorem bnd 6318
Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 4922), derived from the Collection Principle cp 6317. 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.
Assertion
Ref Expression
bnd |- (A.x e. z E.yph -> E.wA.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 6317 . . 3 |- E.wA.x e. z (E.yph -> E.y e. w ph)
2 ralim 2352 . . . 4 |- (A.x e. z (E.yph -> E.y e. w ph) -> (A.x e. z E.yph -> A.x e. z E.y e. w ph))
32eximi 1528 . . 3 |- (E.wA.x e. z (E.yph -> E.y e. w ph) -> E.w(A.x e. z E.yph -> A.x e. z E.y e. w ph))
41, 3ax-mp 7 . 2 |- E.w(A.x e. z E.yph -> A.x e. z E.y e. w ph)
5 19.37v 1884 . 2 |- (E.w(A.x e. z E.yph -> A.x e. z E.y e. w ph) <-> (A.x e. z E.yph -> E.wA.x e. z E.y e. w ph))
64, 5mpbi 231 1 |- (A.x e. z E.yph -> E.wA.x e. z E.y e. w ph)
Colors of variables: wff set class
Syntax hints:   -> wi 3  E.wex 1501  A.wral 2293  E.wrex 2294
This theorem is referenced by:  bnd2 6319
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1497  ax-6 1498  ax-7 1499  ax-gen 1500  ax-8 1577  ax-10 1578  ax-11 1579  ax-12 1580  ax-13 1581  ax-14 1582  ax-17 1589  ax-9 1603  ax-4 1609  ax-16 1786  ax-ext 2057  ax-rep 3582  ax-sep 3592  ax-nul 3602  ax-pow 3638  ax-pr 3662  ax-un 3930  ax-reg 6085  ax-inf2 6120
This theorem depends on definitions:  df-bi 204  df-or 413  df-an 414  df-3or 1019  df-3an 1020  df-ex 1502  df-sb 1748  df-eu 1975  df-mo 1976  df-clab 2063  df-cleq 2068  df-clel 2071  df-ne 2203  df-ral 2297  df-rex 2298  df-rab 2300  df-v 2484  df-sbc 2654  df-csb 2728  df-dif 2787  df-un 2789  df-in 2791  df-ss 2793  df-pss 2795  df-nul 3049  df-if 3149  df-pw 3205  df-sn 3220  df-pr 3221  df-tp 3223  df-op 3224  df-uni 3348  df-int 3382  df-iun 3420  df-iin 3421  df-br 3493  df-opab 3551  df-tr 3566  df-eprel 3745  df-id 3748  df-po 3753  df-so 3765  df-fr 3783  df-we 3799  df-ord 3815  df-on 3816  df-lim 3817  df-suc 3818  df-om 4087  df-xp 4134  df-rel 4135  df-cnv 4136  df-co 4137  df-dm 4138  df-rn 4139  df-res 4140  df-ima 4141  df-fun 4142  df-fn 4143  df-f 4144  df-fv 4148  df-mpt 5185  df-rdg 5443  df-r1 6165  df-rank 6166
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