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Theorem sbmo 2186
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 2080 . . 3  |-  ( [ y  /  x ] E. w A. z (
ph  ->  z  =  w )  <->  E. w [ y  /  x ] A. z ( ph  ->  z  =  w ) )
2 nfv 1609 . . . . . 6  |-  F/ x  z  =  w
32sblim 2021 . . . . 5  |-  ( [ y  /  x ]
( ph  ->  z  =  w )  <->  ( [
y  /  x ] ph  ->  z  =  w ) )
43sbalv 2081 . . . 4  |-  ( [ y  /  x ] A. z ( ph  ->  z  =  w )  <->  A. z
( [ y  /  x ] ph  ->  z  =  w ) )
54exbii 1572 . . 3  |-  ( E. w [ y  /  x ] A. z (
ph  ->  z  =  w )  <->  E. w A. z
( [ y  /  x ] ph  ->  z  =  w ) )
61, 5bitri 240 . 2  |-  ( [ y  /  x ] E. w A. z (
ph  ->  z  =  w )  <->  E. w A. z
( [ y  /  x ] ph  ->  z  =  w ) )
7 nfv 1609 . . . 4  |-  F/ w ph
87mo2 2185 . . 3  |-  ( E* z ph  <->  E. w A. z ( ph  ->  z  =  w ) )
98sbbii 1643 . 2  |-  ( [ y  /  x ] E* z ph  <->  [ y  /  x ] E. w A. z ( ph  ->  z  =  w ) )
10 nfv 1609 . . 3  |-  F/ w [ y  /  x ] ph
1110mo2 2185 . 2  |-  ( E* z [ y  /  x ] ph  <->  E. w A. z ( [ y  /  x ] ph  ->  z  =  w ) )
126, 9, 113bitr4i 268 1  |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   E.wex 1531    = wceq 1632   [wsb 1638   E*wmo 2157
This theorem is referenced by:  sbmoOLD  26442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161
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