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Theorem r19.12sn 3436
Description: Special case of r19.12 2422 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 2836 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph )
)
3 rexsnsOLD 3410 . . 3  |-  ( A  e.  _V  ->  ( E. x  e.  { A } A. y  e.  B  ph  <->  [. A  /  x ]. A. y  e.  B  ph ) )
4 rexsnsOLD 3410 . . . 4  |-  ( A  e.  _V  ->  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
54ralbidv 2326 . . 3  |-  ( A  e.  _V  ->  ( A. y  e.  B  E. x  e.  { A } ph  <->  A. y  e.  B  [. A  /  x ]. ph ) )
62, 3, 53bitr4d 209 . 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 7 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 98    e. wcel 1393   A.wral 2306   E.wrex 2307   _Vcvv 2557   [.wsbc 2764   {csn 3375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-sn 3381
This theorem is referenced by: (None)
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