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Theorem bnd 6228
Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 4811), derived from the Collection Principle cp 6227. 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 6227 . . 3 |- E.wA.x e. z (E.yph -> E.y e. w ph)
2 ralim 2231 . . . 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 1408 . . 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 8 . 2 |- E.w(A.x e. z E.yph -> A.x e. z E.y e. w ph)
5 19.37v 1770 . 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 217 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 4  E.wex 1380  A.wral 2170  E.wrex 2171
This theorem is referenced by:  bnd2 6229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1376  ax-6 1377  ax-7 1378  ax-gen 1379  ax-8 1461  ax-10 1462  ax-11 1463  ax-12 1464  ax-13 1465  ax-14 1466  ax-17 1473  ax-9 1488  ax-4 1494  ax-16 1672  ax-ext 1943  ax-rep 3465  ax-sep 3475  ax-nul 3484  ax-pow 3520  ax-pr 3544  ax-un 3814  ax-reg 6037  ax-inf2 6072
This theorem depends on definitions:  df-bi 190  df-or 383  df-an 384  df-3or 947  df-3an 948  df-ex 1381  df-sb 1634  df-eu 1861  df-mo 1862  df-clab 1949  df-cleq 1954  df-clel 1957  df-ne 2081  df-ral 2174  df-rex 2175  df-rab 2177  df-v 2368  df-sbc 2533  df-csb 2607  df-dif 2666  df-un 2668  df-in 2670  df-ss 2672  df-pss 2674  df-nul 2928  df-if 3029  df-pw 3087  df-sn 3102  df-pr 3103  df-tp 3105  df-op 3106  df-uni 3235  df-int 3269  df-iun 3307  df-iin 3308  df-br 3380  df-opab 3434  df-tr 3449  df-eprel 3627  df-id 3630  df-po 3635  df-so 3649  df-fr 3668  df-we 3684  df-ord 3700  df-on 3701  df-lim 3702  df-suc 3703  df-om 3975  df-xp 4022  df-rel 4023  df-cnv 4024  df-co 4025  df-dm 4026  df-rn 4027  df-res 4028  df-ima 4029  df-fun 4030  df-fn 4031  df-f 4032  df-fv 4036  df-mpt 5072  df-rdg 5359  df-r1 6114  df-rank 6115
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