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Theorem isarep1 5257
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 2715 . . 3  |-  b  e. 
_V
21elima 4934 . 2  |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. z  e.  A  z { <. x ,  y >.  |  ph } b )
3 df-br 3967 . . . 4  |-  ( z { <. x ,  y
>.  |  ph } b  <->  <. z ,  b >.  e.  { <. x ,  y
>.  |  ph } )
4 opelopabsb 4221 . . . 4  |-  ( <.
z ,  b >.  e.  { <. x ,  y
>.  |  ph }  <->  [. z  /  x ]. [. b  / 
y ]. ph )
5 sbsbc 2941 . . . . . 6  |-  ( [ b  /  y ]
ph 
<-> 
[. b  /  y ]. ph )
65sbbii 1745 . . . . 5  |-  ( [ z  /  x ] [ b  /  y ] ph  <->  [ z  /  x ] [. b  /  y ]. ph )
7 sbsbc 2941 . . . . 5  |-  ( [ z  /  x ] [. b  /  y ]. ph  <->  [. z  /  x ]. [. b  /  y ]. ph )
86, 7bitr2i 184 . . . 4  |-  ( [. z  /  x ]. [. b  /  y ]. ph  <->  [ z  /  x ] [ b  /  y ] ph )
93, 4, 83bitri 205 . . 3  |-  ( z { <. x ,  y
>.  |  ph } b  <->  [ z  /  x ] [ b  /  y ] ph )
109rexbii 2464 . 2  |-  ( E. z  e.  A  z { <. x ,  y
>.  |  ph } b  <->  E. z  e.  A  [ z  /  x ] [ b  /  y ] ph )
11 nfs1v 1919 . . 3  |-  F/ x [ z  /  x ] [ b  /  y ] ph
12 nfv 1508 . . 3  |-  F/ z [ b  /  y ] ph
13 sbequ12r 1752 . . 3  |-  ( z  =  x  ->  ( [ z  /  x ] [ b  /  y ] ph  <->  [ b  /  y ] ph ) )
1411, 12, 13cbvrex 2677 . 2  |-  ( E. z  e.  A  [
z  /  x ] [ b  /  y ] ph  <->  E. x  e.  A  [ b  /  y ] ph )
152, 10, 143bitri 205 1  |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. x  e.  A  [ b  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   [wsb 1742    e. wcel 2128   E.wrex 2436   [.wsbc 2937   <.cop 3563   class class class wbr 3966   {copab 4025   "cima 4590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-xp 4593  df-cnv 4595  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600
This theorem is referenced by: (None)
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