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Theorem cp 7037
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 7031 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 2475 . . 3  |-  z  e. 
_V
21cplem2 7036 . 2  |-  E. w A. x  e.  z 
( { y  | 
ph }  =/=  (/)  ->  ( { y  |  ph }  i^i  w )  =/=  (/) )
3 abn0 3087 . . . . 5  |-  ( { y  |  ph }  =/=  (/)  <->  E. y ph )
4 elin 2975 . . . . . . . 8  |-  ( y  e.  ( { y  |  ph }  i^i  w )  <->  ( y  e.  { y  |  ph }  /\  y  e.  w
) )
5 abid 2050 . . . . . . . . 9  |-  ( y  e.  { y  | 
ph }  <->  ph )
65anbi1i 667 . . . . . . . 8  |-  ( ( y  e.  { y  |  ph }  /\  y  e.  w )  <->  (
ph  /\  y  e.  w ) )
7 ancom 430 . . . . . . . 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 1477 . . . . . 6  |-  ( E. y  y  e.  ( { y  |  ph }  i^i  w )  <->  E. y
( y  e.  w  /\  ph ) )
10 hbab1 2051 . . . . . . . 8  |-  ( z  e.  { y  | 
ph }  ->  A. y 
z  e.  { y  |  ph } )
11 ax-17 1526 . . . . . . . 8  |-  ( z  e.  w  ->  A. y 
z  e.  w )
1210, 11hbin 2991 . . . . . . 7  |-  ( z  e.  ( { y  |  ph }  i^i  w )  ->  A. y 
z  e.  ( { y  |  ph }  i^i  w ) )
1312n0f 3077 . . . . . 6  |-  ( ( { y  |  ph }  i^i  w )  =/=  (/) 
<->  E. y  y  e.  ( { y  | 
ph }  i^i  w
) )
14 df-rex 2276 . . . . . 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 2293 . . 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 1477 . 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 1443    e. wcel 1519   {cab 2048    =/= wne 2179   A.wral 2271   E.wrex 2272    i^i cin 2789   (/)c0 3069
This theorem is referenced by:  bnd  7038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1439  ax-6 1440  ax-7 1441  ax-gen 1442  ax-8 1521  ax-11 1522  ax-13 1523  ax-14 1524  ax-17 1526  ax-12o 1559  ax-10 1573  ax-9 1579  ax-4 1586  ax-16 1772  ax-ext 2043  ax-rep 3688  ax-sep 3698  ax-nul 3706  ax-pow 3742  ax-pr 3766  ax-un 4058  ax-reg 6783  ax-inf2 6819
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 894  df-3an 895  df-ex 1444  df-sb 1733  df-eu 1955  df-mo 1956  df-clab 2049  df-cleq 2054  df-clel 2057  df-ne 2181  df-ral 2275  df-rex 2276  df-reu 2277  df-rab 2278  df-v 2474  df-sbc 2648  df-csb 2730  df-dif 2793  df-un 2795  df-in 2797  df-ss 2801  df-pss 2803  df-nul 3070  df-if 3179  df-pw 3240  df-sn 3258  df-pr 3259  df-tp 3260  df-op 3261  df-uni 3422  df-int 3456  df-iun 3499  df-iin 3500  df-br 3584  df-opab 3638  df-mpt 3639  df-tr 3671  df-eprel 3853  df-id 3857  df-po 3862  df-so 3863  df-fr 3900  df-we 3902  df-ord 3943  df-on 3944  df-lim 3945  df-suc 3946  df-om 4221  df-xp 4267  df-rel 4268  df-cnv 4269  df-co 4270  df-dm 4271  df-rn 4272  df-res 4273  df-ima 4274  df-fun 4275  df-fn 4276  df-f 4277  df-f1 4278  df-fo 4279  df-f1o 4280  df-fv 4281  df-recs 5837  df-rdg 5872  df-r1 6913  df-rank 6914
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