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Theorem isarep1 5331
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 2791 . . 3  |-  b  e. 
_V
21elima 5017 . 2  |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. z  e.  A  z { <. x ,  y >.  |  ph } b )
3 df-br 4024 . . . 4  |-  ( z { <. x ,  y
>.  |  ph } b  <->  <. z ,  b >.  e.  { <. x ,  y
>.  |  ph } )
4 opelopabsb 4275 . . . 4  |-  ( <.
z ,  b >.  e.  { <. x ,  y
>.  |  ph }  <->  [. z  /  x ]. [. b  / 
y ]. ph )
5 sbsbc 2995 . . . . . 6  |-  ( [ b  /  y ]
ph 
<-> 
[. b  /  y ]. ph )
65sbbii 1634 . . . . 5  |-  ( [ z  /  x ] [ b  /  y ] ph  <->  [ z  /  x ] [. b  /  y ]. ph )
7 sbsbc 2995 . . . . 5  |-  ( [ z  /  x ] [. b  /  y ]. ph  <->  [. z  /  x ]. [. b  /  y ]. ph )
86, 7bitr2i 241 . . . 4  |-  ( [. z  /  x ]. [. b  /  y ]. ph  <->  [ z  /  x ] [ b  /  y ] ph )
93, 4, 83bitri 262 . . 3  |-  ( z { <. x ,  y
>.  |  ph } b  <->  [ z  /  x ] [ b  /  y ] ph )
109rexbii 2568 . 2  |-  ( E. z  e.  A  z { <. x ,  y
>.  |  ph } b  <->  E. z  e.  A  [ z  /  x ] [ b  /  y ] ph )
11 nfs1v 2045 . . 3  |-  F/ x [ z  /  x ] [ b  /  y ] ph
12 nfv 1605 . . 3  |-  F/ z [ b  /  y ] ph
13 sbequ12r 1861 . . 3  |-  ( z  =  x  ->  ( [ z  /  x ] [ b  /  y ] ph  <->  [ b  /  y ] ph ) )
1411, 12, 13cbvrex 2761 . 2  |-  ( E. z  e.  A  [
z  /  x ] [ b  /  y ] ph  <->  E. x  e.  A  [ b  /  y ] ph )
152, 10, 143bitri 262 1  |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. x  e.  A  [ b  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [wsb 1629    e. wcel 1684   E.wrex 2544   [.wsbc 2991   <.cop 3643   class class class wbr 4023   {copab 4076   "cima 4692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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