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Theorem sbmo 2354
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 2302 . . 3  |-  ( [ y  /  x ] E. w A. z (
ph  ->  z  =  w )  <->  E. w [ y  /  x ] A. z ( ph  ->  z  =  w ) )
2 nfv 1771 . . . . . 6  |-  F/ x  z  =  w
32sblim 2236 . . . . 5  |-  ( [ y  /  x ]
( ph  ->  z  =  w )  <->  ( [
y  /  x ] ph  ->  z  =  w ) )
43sbalv 2303 . . . 4  |-  ( [ y  /  x ] A. z ( ph  ->  z  =  w )  <->  A. z
( [ y  /  x ] ph  ->  z  =  w ) )
54exbii 1728 . . 3  |-  ( E. w [ y  /  x ] A. z (
ph  ->  z  =  w )  <->  E. w A. z
( [ y  /  x ] ph  ->  z  =  w ) )
61, 5bitri 257 . 2  |-  ( [ y  /  x ] E. w A. z (
ph  ->  z  =  w )  <->  E. w A. z
( [ y  /  x ] ph  ->  z  =  w ) )
7 mo2v 2316 . . 3  |-  ( E* z ph  <->  E. w A. z ( ph  ->  z  =  w ) )
87sbbii 1814 . 2  |-  ( [ y  /  x ] E* z ph  <->  [ y  /  x ] E. w A. z ( ph  ->  z  =  w ) )
9 mo2v 2316 . 2  |-  ( E* z [ y  /  x ] ph  <->  E. w A. z ( [ y  /  x ] ph  ->  z  =  w ) )
106, 8, 93bitr4i 285 1  |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1452   E.wex 1673   [wsb 1807   E*wmo 2310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator