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Theorem isarep1 5532
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 2959 . . 3  |-  b  e. 
_V
21elima 5208 . 2  |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. z  e.  A  z { <. x ,  y >.  |  ph } b )
3 df-br 4213 . . . 4  |-  ( z { <. x ,  y
>.  |  ph } b  <->  <. z ,  b >.  e.  { <. x ,  y
>.  |  ph } )
4 opelopabsb 4465 . . . 4  |-  ( <.
z ,  b >.  e.  { <. x ,  y
>.  |  ph }  <->  [. z  /  x ]. [. b  / 
y ]. ph )
5 sbsbc 3165 . . . . . 6  |-  ( [ b  /  y ]
ph 
<-> 
[. b  /  y ]. ph )
65sbbii 1665 . . . . 5  |-  ( [ z  /  x ] [ b  /  y ] ph  <->  [ z  /  x ] [. b  /  y ]. ph )
7 sbsbc 3165 . . . . 5  |-  ( [ z  /  x ] [. b  /  y ]. ph  <->  [. z  /  x ]. [. b  /  y ]. ph )
86, 7bitr2i 242 . . . 4  |-  ( [. z  /  x ]. [. b  /  y ]. ph  <->  [ z  /  x ] [ b  /  y ] ph )
93, 4, 83bitri 263 . . 3  |-  ( z { <. x ,  y
>.  |  ph } b  <->  [ z  /  x ] [ b  /  y ] ph )
109rexbii 2730 . 2  |-  ( E. z  e.  A  z { <. x ,  y
>.  |  ph } b  <->  E. z  e.  A  [ z  /  x ] [ b  /  y ] ph )
11 nfs1v 2182 . . 3  |-  F/ x [ z  /  x ] [ b  /  y ] ph
12 nfv 1629 . . 3  |-  F/ z [ b  /  y ] ph
13 sbequ12r 1945 . . 3  |-  ( z  =  x  ->  ( [ z  /  x ] [ b  /  y ] ph  <->  [ b  /  y ] ph ) )
1411, 12, 13cbvrex 2929 . 2  |-  ( E. z  e.  A  [
z  /  x ] [ b  /  y ] ph  <->  E. x  e.  A  [ b  /  y ] ph )
152, 10, 143bitri 263 1  |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. x  e.  A  [ b  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   [wsb 1658    e. wcel 1725   E.wrex 2706   [.wsbc 3161   <.cop 3817   class class class wbr 4212   {copab 4265   "cima 4881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891
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