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Theorem r19.12sn 3874
Description: Special case of r19.12 2821 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
r19.12sn.1  |-  A  e. 
_V
Assertion
Ref Expression
r19.12sn  |-  ( 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)

Proof of Theorem r19.12sn
StepHypRef Expression
1 r19.12sn.1 . 2  |-  A  e. 
_V
2 sbcralg 3237 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph )
)
3 rexsns 3847 . . 3  |-  ( A  e.  _V  ->  ( E. x  e.  { A } A. y  e.  B  ph  <->  [. A  /  x ]. A. y  e.  B  ph ) )
4 rexsns 3847 . . . 4  |-  ( A  e.  _V  ->  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
54ralbidv 2727 . . 3  |-  ( A  e.  _V  ->  ( A. y  e.  B  E. x  e.  { A } ph  <->  A. y  e.  B  [. A  /  x ]. ph ) )
62, 3, 53bitr4d 278 . 2  |-  ( A  e.  _V  ->  ( E. x  e.  { A } A. y  e.  B  ph  <->  A. y  e.  B  E. x  e.  { A } ph ) )
71, 6ax-mp 8 1  |-  ( 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:    <-> wb 178    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958   [.wsbc 3163   {csn 3816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-sbc 3164  df-sn 3822
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