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Theorem isarep1 5673
Description: Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by  ph ( x ,  y ) i.e. the class  ( { <. x ,  y >.  |  ph } " A ). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
isarep1  |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. x  e.  A  [ b  /  y ] ph )
Distinct variable groups:    x, A    x, b, y
Allowed substitution hints:    ph( x, y, b)    A( y, b)

Proof of Theorem isarep1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 vex 3121 . . 3  |-  b  e. 
_V
21elima 5348 . 2  |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. z  e.  A  z { <. x ,  y >.  |  ph } b )
3 df-br 4454 . . . 4  |-  ( z { <. x ,  y
>.  |  ph } b  <->  <. z ,  b >.  e.  { <. x ,  y
>.  |  ph } )
4 opelopabsb 4763 . . . 4  |-  ( <.
z ,  b >.  e.  { <. x ,  y
>.  |  ph }  <->  [. z  /  x ]. [. b  / 
y ]. ph )
5 sbsbc 3340 . . . . . 6  |-  ( [ b  /  y ]
ph 
<-> 
[. b  /  y ]. ph )
65sbbii 1718 . . . . 5  |-  ( [ z  /  x ] [ b  /  y ] ph  <->  [ z  /  x ] [. b  /  y ]. ph )
7 sbsbc 3340 . . . . 5  |-  ( [ z  /  x ] [. b  /  y ]. ph  <->  [. z  /  x ]. [. b  /  y ]. ph )
86, 7bitr2i 250 . . . 4  |-  ( [. z  /  x ]. [. b  /  y ]. ph  <->  [ z  /  x ] [ b  /  y ] ph )
93, 4, 83bitri 271 . . 3  |-  ( z { <. x ,  y
>.  |  ph } b  <->  [ z  /  x ] [ b  /  y ] ph )
109rexbii 2969 . 2  |-  ( E. z  e.  A  z { <. x ,  y
>.  |  ph } b  <->  E. z  e.  A  [ z  /  x ] [ b  /  y ] ph )
11 nfs1v 2164 . . 3  |-  F/ x [ z  /  x ] [ b  /  y ] ph
12 nfv 1683 . . 3  |-  F/ z [ b  /  y ] ph
13 sbequ12r 1962 . . 3  |-  ( z  =  x  ->  ( [ z  /  x ] [ b  /  y ] ph  <->  [ b  /  y ] ph ) )
1411, 12, 13cbvrex 3090 . 2  |-  ( E. z  e.  A  [
z  /  x ] [ b  /  y ] ph  <->  E. x  e.  A  [ b  /  y ] ph )
152, 10, 143bitri 271 1  |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. x  e.  A  [ b  /  y ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   [wsb 1711    e. wcel 1767   E.wrex 2818   [.wsbc 3336   <.cop 4039   class class class wbr 4453   {copab 4510   "cima 5008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-cnv 5013  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018
This theorem is referenced by: (None)
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