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Theorem cp 7043
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 7037 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 2478 . . 3  |-  z  e. 
_V
21cplem2 7042 . 2  |-  E. w A. x  e.  z 
( { y  | 
ph }  =/=  (/)  ->  ( { y  |  ph }  i^i  w )  =/=  (/) )
3 abn0 3090 . . . . 5  |-  ( { y  |  ph }  =/=  (/)  <->  E. y ph )
4 elin 2978 . . . . . . . 8  |-  ( y  e.  ( { y  |  ph }  i^i  w )  <->  ( y  e.  { y  |  ph }  /\  y  e.  w
) )
5 abid 2053 . . . . . . . . 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 2054 . . . . . . . 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 2994 . . . . . . 7  |-  ( z  e.  ( { y  |  ph }  i^i  w )  ->  A. y 
z  e.  ( { y  |  ph }  i^i  w ) )
1312n0f 3080 . . . . . 6  |-  ( ( { y  |  ph }  i^i  w )  =/=  (/) 
<->  E. y  y  e.  ( { y  | 
ph }  i^i  w
) )
14 df-rex 2279 . . . . . 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 2296 . . 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 2051    =/= wne 2182   A.wral 2274   E.wrex 2275    i^i cin 2792   (/)c0 3072
This theorem is referenced by:  bnd  7044
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
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