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Theorem moanim 2016
Description: Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)
Hypothesis
Ref Expression
moanim.1  |-  F/ x ph
Assertion
Ref Expression
moanim  |-  ( E* x ( ph  /\  ps )  <->  ( ph  ->  E* x ps ) )

Proof of Theorem moanim
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 anandi 555 . . . . 5  |-  ( (
ph  /\  ( ps  /\ 
[ y  /  x ] ps ) )  <->  ( ( ph  /\  ps )  /\  ( ph  /\  [ y  /  x ] ps ) ) )
21imbi1i 236 . . . 4  |-  ( ( ( ph  /\  ( ps  /\  [ y  /  x ] ps ) )  ->  x  =  y )  <->  ( ( (
ph  /\  ps )  /\  ( ph  /\  [
y  /  x ] ps ) )  ->  x  =  y ) )
3 impexp 259 . . . 4  |-  ( ( ( ph  /\  ( ps  /\  [ y  /  x ] ps ) )  ->  x  =  y )  <->  ( ph  ->  ( ( ps  /\  [
y  /  x ] ps )  ->  x  =  y ) ) )
4 sban 1871 . . . . . . 7  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
5 moanim.1 . . . . . . . . 9  |-  F/ x ph
65sbf 1701 . . . . . . . 8  |-  ( [ y  /  x ] ph 
<-> 
ph )
76anbi1i 446 . . . . . . 7  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) 
<->  ( ph  /\  [
y  /  x ] ps ) )
84, 7bitr2i 183 . . . . . 6  |-  ( (
ph  /\  [ y  /  x ] ps )  <->  [ y  /  x ]
( ph  /\  ps )
)
98anbi2i 445 . . . . 5  |-  ( ( ( ph  /\  ps )  /\  ( ph  /\  [ y  /  x ] ps ) )  <->  ( ( ph  /\  ps )  /\  [ y  /  x ]
( ph  /\  ps )
) )
109imbi1i 236 . . . 4  |-  ( ( ( ( ph  /\  ps )  /\  ( ph  /\  [ y  /  x ] ps ) )  ->  x  =  y )  <->  ( ( (
ph  /\  ps )  /\  [ y  /  x ] ( ph  /\  ps ) )  ->  x  =  y ) )
112, 3, 103bitr3i 208 . . 3  |-  ( (
ph  ->  ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
)  <->  ( ( (
ph  /\  ps )  /\  [ y  /  x ] ( ph  /\  ps ) )  ->  x  =  y ) )
12112albii 1401 . 2  |-  ( A. x A. y ( ph  ->  ( ( ps  /\  [ y  /  x ] ps )  ->  x  =  y ) )  <->  A. x A. y ( ( (
ph  /\  ps )  /\  [ y  /  x ] ( ph  /\  ps ) )  ->  x  =  y ) )
13519.21 1516 . . 3  |-  ( A. x ( ph  ->  A. y ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
)  <->  ( ph  ->  A. x A. y ( ( ps  /\  [
y  /  x ] ps )  ->  x  =  y ) ) )
14 19.21v 1795 . . . 4  |-  ( A. y ( ph  ->  ( ( ps  /\  [
y  /  x ] ps )  ->  x  =  y ) )  <->  ( ph  ->  A. y ( ( ps  /\  [ y  /  x ] ps )  ->  x  =  y ) ) )
1514albii 1400 . . 3  |-  ( A. x A. y ( ph  ->  ( ( ps  /\  [ y  /  x ] ps )  ->  x  =  y ) )  <->  A. x
( ph  ->  A. y
( ( ps  /\  [ y  /  x ] ps )  ->  x  =  y ) ) )
16 ax-17 1460 . . . . 5  |-  ( ps 
->  A. y ps )
1716mo3h 1995 . . . 4  |-  ( E* x ps  <->  A. x A. y ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
)
1817imbi2i 224 . . 3  |-  ( (
ph  ->  E* x ps )  <->  ( ph  ->  A. x A. y ( ( ps  /\  [
y  /  x ] ps )  ->  x  =  y ) ) )
1913, 15, 183bitr4ri 211 . 2  |-  ( (
ph  ->  E* x ps )  <->  A. x A. y
( ph  ->  ( ( ps  /\  [ y  /  x ] ps )  ->  x  =  y ) ) )
20 ax-17 1460 . . 3  |-  ( (
ph  /\  ps )  ->  A. y ( ph  /\ 
ps ) )
2120mo3h 1995 . 2  |-  ( E* x ( ph  /\  ps )  <->  A. x A. y
( ( ( ph  /\ 
ps )  /\  [
y  /  x ]
( ph  /\  ps )
)  ->  x  =  y ) )
2212, 19, 213bitr4ri 211 1  |-  ( E* x ( ph  /\  ps )  <->  ( ph  ->  E* x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283   F/wnf 1390   [wsb 1686   E*wmo 1943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946
This theorem is referenced by:  moanimv  2017  moaneu  2018  moanmo  2019
  Copyright terms: Public domain W3C validator