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Theorem sbmo 2313
Description: Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
sbmo  |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
Distinct variable groups:    x, z    y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sbmo
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 sbex 2207 . . 3  |-  ( [ y  /  x ] E. w A. z (
ph  ->  z  =  w )  <->  E. w [ y  /  x ] A. z ( ph  ->  z  =  w ) )
2 nfv 1630 . . . . . 6  |-  F/ x  z  =  w
32sblim 2139 . . . . 5  |-  ( [ y  /  x ]
( ph  ->  z  =  w )  <->  ( [
y  /  x ] ph  ->  z  =  w ) )
43sbalv 2208 . . . 4  |-  ( [ y  /  x ] A. z ( ph  ->  z  =  w )  <->  A. z
( [ y  /  x ] ph  ->  z  =  w ) )
54exbii 1593 . . 3  |-  ( E. w [ y  /  x ] A. z (
ph  ->  z  =  w )  <->  E. w A. z
( [ y  /  x ] ph  ->  z  =  w ) )
61, 5bitri 242 . 2  |-  ( [ y  /  x ] E. w A. z (
ph  ->  z  =  w )  <->  E. w A. z
( [ y  /  x ] ph  ->  z  =  w ) )
7 nfv 1630 . . . 4  |-  F/ w ph
87mo2 2312 . . 3  |-  ( E* z ph  <->  E. w A. z ( ph  ->  z  =  w ) )
98sbbii 1666 . 2  |-  ( [ y  /  x ] E* z ph  <->  [ y  /  x ] E. w A. z ( ph  ->  z  =  w ) )
10 nfv 1630 . . 3  |-  F/ w [ y  /  x ] ph
1110mo2 2312 . 2  |-  ( E* z [ y  /  x ] ph  <->  E. w A. z ( [ y  /  x ] ph  ->  z  =  w ) )
126, 9, 113bitr4i 270 1  |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550   E.wex 1551   [wsb 1659   E*wmo 2284
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
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-eu 2287  df-mo 2288
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