HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version

Theorem bnd 7044
Description: A very strong generalization of the Axiom of Replacement (compare zfrep6 5197), derived from the Collection Principle cp 7043. 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 7043 . . 3  |-  E. w A. x  e.  z 
( E. y ph  ->  E. y  e.  w  ph )
2 ralim 2336 . . . 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 1474 . . 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 1877 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 1446   A.wral 2274   E.wrex 2275
This theorem is referenced by:  bnd2  7045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1442  ax-6 1443  ax-7 1444  ax-gen 1445  ax-8 1524  ax-11 1525  ax-13 1526  ax-14 1527  ax-17 1529  ax-12o 1562  ax-10 1576  ax-9 1582  ax-4 1589  ax-16 1775  ax-ext 2046  ax-rep 3691  ax-sep 3701  ax-nul 3709  ax-pow 3745  ax-pr 3769  ax-un 4061  ax-reg 6789  ax-inf2 6825
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 897  df-3an 898  df-ex 1447  df-sb 1736  df-eu 1958  df-mo 1959  df-clab 2052  df-cleq 2057  df-clel 2060  df-ne 2184  df-ral 2278  df-rex 2279  df-reu 2280  df-rab 2281  df-v 2477  df-sbc 2651  df-csb 2733  df-dif 2796  df-un 2798  df-in 2800  df-ss 2804  df-pss 2806  df-nul 3073  df-if 3182  df-pw 3243  df-sn 3261  df-pr 3262  df-tp 3263  df-op 3264  df-uni 3425  df-int 3459  df-iun 3502  df-iin 3503  df-br 3587  df-opab 3641  df-mpt 3642  df-tr 3674  df-eprel 3856  df-id 3860  df-po 3865  df-so 3866  df-fr 3903  df-we 3905  df-ord 3946  df-on 3947  df-lim 3948  df-suc 3949  df-om 4224  df-xp 4270  df-rel 4271  df-cnv 4272  df-co 4273  df-dm 4274  df-rn 4275  df-res 4276  df-ima 4277  df-fun 4278  df-fn 4279  df-f 4280  df-f1 4281  df-fo 4282  df-f1o 4283  df-fv 4284  df-recs 5843  df-rdg 5878  df-r1 6919  df-rank 6920
Copyright terms: Public domain