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Theorem cp 7049
Description: Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 7043 that collapses a proper class into a set of minimum rank. The wff  ph can be thought of as  ph ( x ,  y ). Scheme "Collection Principle" of [Jech] p. 72. (Contributed by NM, 17-Oct-2003.)
Assertion
Ref Expression
cp  |-  E. w A. x  e.  z 
( E. y ph  ->  E. y  e.  w  ph )
Distinct variable groups:    ph, z, w   
x, y, z, w
Allowed substitution hints:    ph( x, y)

Proof of Theorem cp
StepHypRef Expression
1 vex 2479 . . 3  |-  z  e. 
_V
21cplem2 7048 . 2  |-  E. w A. x  e.  z 
( { y  | 
ph }  =/=  (/)  ->  ( { y  |  ph }  i^i  w )  =/=  (/) )
3 abn0 3091 . . . . 5  |-  ( { y  |  ph }  =/=  (/)  <->  E. y ph )
4 elin 2979 . . . . . . . 8  |-  ( y  e.  ( { y  |  ph }  i^i  w )  <->  ( y  e.  { y  |  ph }  /\  y  e.  w
) )
5 abid 2054 . . . . . . . . 9  |-  ( y  e.  { y  | 
ph }  <->  ph )
65anbi1i 669 . . . . . . . 8  |-  ( ( y  e.  { y  |  ph }  /\  y  e.  w )  <->  (
ph  /\  y  e.  w ) )
7 ancom 431 . . . . . . . 8  |-  ( (
ph  /\  y  e.  w )  <->  ( y  e.  w  /\  ph )
)
84, 6, 73bitri 260 . . . . . . 7  |-  ( y  e.  ( { y  |  ph }  i^i  w )  <->  ( y  e.  w  /\  ph )
)
98exbii 1480 . . . . . 6  |-  ( E. y  y  e.  ( { y  |  ph }  i^i  w )  <->  E. y
( y  e.  w  /\  ph ) )
10 hbab1 2055 . . . . . . . 8  |-  ( z  e.  { y  | 
ph }  ->  A. y 
z  e.  { y  |  ph } )
11 ax-17 1529 . . . . . . . 8  |-  ( z  e.  w  ->  A. y 
z  e.  w )
1210, 11hbin 2995 . . . . . . 7  |-  ( z  e.  ( { y  |  ph }  i^i  w )  ->  A. y 
z  e.  ( { y  |  ph }  i^i  w ) )
1312n0f 3081 . . . . . 6  |-  ( ( { y  |  ph }  i^i  w )  =/=  (/) 
<->  E. y  y  e.  ( { y  | 
ph }  i^i  w
) )
14 df-rex 2280 . . . . . 6  |-  ( E. y  e.  w  ph  <->  E. y ( y  e.  w  /\  ph )
)
159, 13, 143bitr4i 266 . . . . 5  |-  ( ( { y  |  ph }  i^i  w )  =/=  (/) 
<->  E. y  e.  w  ph )
163, 15imbi12i 314 . . . 4  |-  ( ( { y  |  ph }  =/=  (/)  ->  ( {
y  |  ph }  i^i  w )  =/=  (/) )  <->  ( E. y ph  ->  E. y  e.  w  ph ) )
1716ralbii 2297 . . 3  |-  ( A. x  e.  z  ( { y  |  ph }  =/=  (/)  ->  ( {
y  |  ph }  i^i  w )  =/=  (/) )  <->  A. x  e.  z  ( E. y ph  ->  E. y  e.  w  ph ) )
1817exbii 1480 . 2  |-  ( E. w A. x  e.  z  ( { y  |  ph }  =/=  (/) 
->  ( { y  | 
ph }  i^i  w
)  =/=  (/) )  <->  E. w A. x  e.  z 
( E. y ph  ->  E. y  e.  w  ph ) )
192, 18mpbi 197 1  |-  E. w A. x  e.  z 
( E. y ph  ->  E. y  e.  w  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 356   E.wex 1446    e. wcel 1522   {cab 2052    =/= wne 2183   A.wral 2275   E.wrex 2276    i^i cin 2793   (/)c0 3073
This theorem is referenced by:  bnd  7050
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 1563  ax-10 1577  ax-9 1583  ax-4 1590  ax-16 1776  ax-ext 2047  ax-rep 3697  ax-sep 3707  ax-nul 3715  ax-pow 3751  ax-pr 3775  ax-un 4067  ax-reg 6795  ax-inf2 6831
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 1737  df-eu 1959  df-mo 1960  df-clab 2053  df-cleq 2058  df-clel 2061  df-ne 2185  df-ral 2279  df-rex 2280  df-reu 2281  df-rab 2282  df-v 2478  df-sbc 2652  df-csb 2734  df-dif 2797  df-un 2799  df-in 2801  df-ss 2805  df-pss 2807  df-nul 3074  df-if 3183  df-pw 3244  df-sn 3262  df-pr 3263  df-tp 3264  df-op 3265  df-uni 3431  df-int 3465  df-iun 3508  df-iin 3509  df-br 3593  df-opab 3647  df-mpt 3648  df-tr 3680  df-eprel 3862  df-id 3866  df-po 3871  df-so 3872  df-fr 3909  df-we 3911  df-ord 3952  df-on 3953  df-lim 3954  df-suc 3955  df-om 4230  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-res 4282  df-ima 4283  df-fun 4284  df-fn 4285  df-f 4286  df-f1 4287  df-fo 4288  df-f1o 4289  df-fv 4290  df-recs 5849  df-rdg 5884  df-r1 6925  df-rank 6926
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