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Theorem sbcreu 3411
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 3334 . 2  |-  ( [. A  /  x ]. E! y  e.  B  ph  ->  A  e.  _V )
2 reurex 3071 . . 3  |-  ( E! y  e.  B  [. A  /  x ]. ph  ->  E. y  e.  B  [. A  /  x ]. ph )
3 sbcex 3334 . . . 4  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
43rexlimivw 2945 . . 3  |-  ( E. y  e.  B  [. A  /  x ]. ph  ->  A  e.  _V )
52, 4syl 16 . 2  |-  ( E! y  e.  B  [. A  /  x ]. ph  ->  A  e.  _V )
6 dfsbcq2 3327 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] E! y  e.  B  ph  <->  [. A  /  x ]. E! y  e.  B  ph ) )
7 dfsbcq2 3327 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
87reubidv 3039 . . 3  |-  ( z  =  A  ->  ( E! y  e.  B  [ z  /  x ] ph  <->  E! y  e.  B  [. A  /  x ]. ph ) )
9 nfcv 2622 . . . . 5  |-  F/_ x B
10 nfs1v 2157 . . . . 5  |-  F/ x [ z  /  x ] ph
119, 10nfreu 3029 . . . 4  |-  F/ x E! y  e.  B  [ z  /  x ] ph
12 sbequ12 1954 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
1312reubidv 3039 . . . 4  |-  ( x  =  z  ->  ( E! y  e.  B  ph  <->  E! y  e.  B  [
z  /  x ] ph ) )
1411, 13sbie 2116 . . 3  |-  ( [ z  /  x ] E! y  e.  B  ph  <->  E! y  e.  B  [
z  /  x ] ph )
156, 8, 14vtoclbg 3165 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. E! y  e.  B  ph  <->  E! y  e.  B  [. A  /  x ]. ph )
)
161, 5, 15pm5.21nii 353 1  |-  ( [. A  /  x ]. E! y  e.  B  ph  <->  E! y  e.  B  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1374   [wsb 1706    e. wcel 1762   E.wrex 2808   E!wreu 2809   _Vcvv 3106   [.wsbc 3324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-v 3108  df-sbc 3325
This theorem is referenced by: (None)
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