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Theorem sbcreu 3344
Description: Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcreu  |-  ( [. A  /  x ]. E! y  e.  B  ph  <->  E! y  e.  B  [. A  /  x ]. ph )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)

Proof of Theorem sbcreu
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbcex 3277 . 2  |-  ( [. A  /  x ]. E! y  e.  B  ph  ->  A  e.  _V )
2 reurex 3009 . . 3  |-  ( E! y  e.  B  [. A  /  x ]. ph  ->  E. y  e.  B  [. A  /  x ]. ph )
3 sbcex 3277 . . . 4  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
43rexlimivw 2876 . . 3  |-  ( E. y  e.  B  [. A  /  x ]. ph  ->  A  e.  _V )
52, 4syl 17 . 2  |-  ( E! y  e.  B  [. A  /  x ]. ph  ->  A  e.  _V )
6 dfsbcq2 3270 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] E! y  e.  B  ph  <->  [. A  /  x ]. E! y  e.  B  ph ) )
7 dfsbcq2 3270 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
87reubidv 2975 . . 3  |-  ( z  =  A  ->  ( E! y  e.  B  [ z  /  x ] ph  <->  E! y  e.  B  [. A  /  x ]. ph ) )
9 nfcv 2592 . . . . 5  |-  F/_ x B
10 nfs1v 2266 . . . . 5  |-  F/ x [ z  /  x ] ph
119, 10nfreu 2965 . . . 4  |-  F/ x E! y  e.  B  [ z  /  x ] ph
12 sbequ12 2083 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
1312reubidv 2975 . . . 4  |-  ( x  =  z  ->  ( E! y  e.  B  ph  <->  E! y  e.  B  [
z  /  x ] ph ) )
1411, 13sbie 2237 . . 3  |-  ( [ z  /  x ] E! y  e.  B  ph  <->  E! y  e.  B  [
z  /  x ] ph )
156, 8, 14vtoclbg 3108 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. E! y  e.  B  ph  <->  E! y  e.  B  [. A  /  x ]. ph )
)
161, 5, 15pm5.21nii 355 1  |-  ( [. A  /  x ]. E! y  e.  B  ph  <->  E! y  e.  B  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1444   [wsb 1797    e. wcel 1887   E.wrex 2738   E!wreu 2739   _Vcvv 3045   [.wsbc 3267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-v 3047  df-sbc 3268
This theorem is referenced by: (None)
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