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Theorem r19.12sn 3696
Description: Special case of r19.12 2656 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 3065 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph )
)
3 rexsns 3671 . . 3  |-  ( A  e.  _V  ->  ( E. x  e.  { A } A. y  e.  B  ph  <->  [. A  /  x ]. A. y  e.  B  ph ) )
4 rexsns 3671 . . . 4  |-  ( A  e.  _V  ->  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
54ralbidv 2563 . . 3  |-  ( A  e.  _V  ->  ( A. y  e.  B  E. x  e.  { A } ph  <->  A. y  e.  B  [. A  /  x ]. ph ) )
62, 3, 53bitr4d 276 . 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 176    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788   [.wsbc 2991   {csn 3640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-sn 3646
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