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Theorem r19.12sn 3508
Description: Special case of r19.12 2478 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 20-Dec-2021.)
Assertion
Ref Expression
r19.12sn  |-  ( A  e.  V  ->  ( E. x  e.  { A } A. y  e.  B  ph  <->  A. y  e.  B  E. x  e.  { A } ph ) )
Distinct variable groups:    x, y, A   
x, B
Allowed substitution hints:    ph( x, y)    B( y)    V( x, y)

Proof of Theorem r19.12sn
StepHypRef Expression
1 sbcralg 2917 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph )
)
2 rexsns 3482 . 2  |-  ( E. x  e.  { A } A. y  e.  B  ph  <->  [. A  /  x ]. A. y  e.  B  ph )
3 rexsns 3482 . . 3  |-  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph )
43ralbii 2384 . 2  |-  ( A. y  e.  B  E. x  e.  { A } ph  <->  A. y  e.  B  [. A  /  x ]. ph )
51, 2, 43bitr4g 221 1  |-  ( A  e.  V  ->  ( E. x  e.  { A } A. y  e.  B  ph  <->  A. y  e.  B  E. x  e.  { A } ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1438   A.wral 2359   E.wrex 2360   [.wsbc 2840   {csn 3446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-sn 3452
This theorem is referenced by: (None)
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