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Theorem moanim 2071
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 579 . . . . 5  |-  ( (
ph  /\  ( ps  /\ 
[ y  /  x ] ps ) )  <->  ( ( ph  /\  ps )  /\  ( ph  /\  [ y  /  x ] ps ) ) )
21imbi1i 237 . . . 4  |-  ( ( ( ph  /\  ( ps  /\  [ y  /  x ] ps ) )  ->  x  =  y )  <->  ( ( (
ph  /\  ps )  /\  ( ph  /\  [
y  /  x ] ps ) )  ->  x  =  y ) )
3 impexp 261 . . . 4  |-  ( ( ( ph  /\  ( ps  /\  [ y  /  x ] ps ) )  ->  x  =  y )  <->  ( ph  ->  ( ( ps  /\  [
y  /  x ] ps )  ->  x  =  y ) ) )
4 sban 1926 . . . . . . 7  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
5 moanim.1 . . . . . . . . 9  |-  F/ x ph
65sbf 1750 . . . . . . . 8  |-  ( [ y  /  x ] ph 
<-> 
ph )
76anbi1i 453 . . . . . . 7  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) 
<->  ( ph  /\  [
y  /  x ] ps ) )
84, 7bitr2i 184 . . . . . 6  |-  ( (
ph  /\  [ y  /  x ] ps )  <->  [ y  /  x ]
( ph  /\  ps )
)
98anbi2i 452 . . . . 5  |-  ( ( ( ph  /\  ps )  /\  ( ph  /\  [ y  /  x ] ps ) )  <->  ( ( ph  /\  ps )  /\  [ y  /  x ]
( ph  /\  ps )
) )
109imbi1i 237 . . . 4  |-  ( ( ( ( ph  /\  ps )  /\  ( ph  /\  [ y  /  x ] ps ) )  ->  x  =  y )  <->  ( ( (
ph  /\  ps )  /\  [ y  /  x ] ( ph  /\  ps ) )  ->  x  =  y ) )
112, 3, 103bitr3i 209 . . 3  |-  ( (
ph  ->  ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
)  <->  ( ( (
ph  /\  ps )  /\  [ y  /  x ] ( ph  /\  ps ) )  ->  x  =  y ) )
12112albii 1447 . 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 1562 . . 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 1845 . . . 4  |-  ( A. y ( ph  ->  ( ( ps  /\  [
y  /  x ] ps )  ->  x  =  y ) )  <->  ( ph  ->  A. y ( ( ps  /\  [ y  /  x ] ps )  ->  x  =  y ) ) )
1514albii 1446 . . 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 1506 . . . . 5  |-  ( ps 
->  A. y ps )
1716mo3h 2050 . . . 4  |-  ( E* x ps  <->  A. x A. y ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
)
1817imbi2i 225 . . 3  |-  ( (
ph  ->  E* x ps )  <->  ( ph  ->  A. x A. y ( ( ps  /\  [
y  /  x ] ps )  ->  x  =  y ) ) )
1913, 15, 183bitr4ri 212 . 2  |-  ( (
ph  ->  E* x ps )  <->  A. x A. y
( ph  ->  ( ( ps  /\  [ y  /  x ] ps )  ->  x  =  y ) ) )
20 ax-17 1506 . . 3  |-  ( (
ph  /\  ps )  ->  A. y ( ph  /\ 
ps ) )
2120mo3h 2050 . 2  |-  ( E* x ( ph  /\  ps )  <->  A. x A. y
( ( ( ph  /\ 
ps )  /\  [
y  /  x ]
( ph  /\  ps )
)  ->  x  =  y ) )
2212, 19, 213bitr4ri 212 1  |-  ( E* x ( ph  /\  ps )  <->  ( ph  ->  E* x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1329   F/wnf 1436   [wsb 1735   E*wmo 1998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001
This theorem is referenced by:  moanimv  2072  moaneu  2073  moanmo  2074
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