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Theorem r19.12sn 3559
Description: Special case of r19.12 2515 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 2959 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph )
)
2 rexsns 3533 . 2  |-  ( E. x  e.  { A } A. y  e.  B  ph  <->  [. A  /  x ]. A. y  e.  B  ph )
3 rexsns 3533 . . 3  |-  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph )
43ralbii 2418 . 2  |-  ( A. y  e.  B  E. x  e.  { A } ph  <->  A. y  e.  B  [. A  /  x ]. ph )
51, 2, 43bitr4g 222 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 104    e. wcel 1465   A.wral 2393   E.wrex 2394   [.wsbc 2882   {csn 3497
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-sn 3503
This theorem is referenced by: (None)
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