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Theorem iotaval 5207
Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaval  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem iotaval
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfiota2 5197 . 2  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
2 vex 2755 . . . . . . 7  |-  y  e. 
_V
3 sbeqalb 3034 . . . . . . . 8  |-  ( y  e.  _V  ->  (
( A. x (
ph 
<->  x  =  y )  /\  A. x (
ph 
<->  x  =  z ) )  ->  y  =  z ) )
4 equcomi 1715 . . . . . . . 8  |-  ( y  =  z  ->  z  =  y )
53, 4syl6 33 . . . . . . 7  |-  ( y  e.  _V  ->  (
( A. x (
ph 
<->  x  =  y )  /\  A. x (
ph 
<->  x  =  z ) )  ->  z  =  y ) )
62, 5ax-mp 5 . . . . . 6  |-  ( ( A. x ( ph  <->  x  =  y )  /\  A. x ( ph  <->  x  =  z ) )  -> 
z  =  y )
76ex 115 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  ( A. x ( ph  <->  x  =  z )  ->  z  =  y ) )
8 equequ2 1724 . . . . . . . . . 10  |-  ( y  =  z  ->  (
x  =  y  <->  x  =  z ) )
98equcoms 1719 . . . . . . . . 9  |-  ( z  =  y  ->  (
x  =  y  <->  x  =  z ) )
109bibi2d 232 . . . . . . . 8  |-  ( z  =  y  ->  (
( ph  <->  x  =  y
)  <->  ( ph  <->  x  =  z ) ) )
1110biimpd 144 . . . . . . 7  |-  ( z  =  y  ->  (
( ph  <->  x  =  y
)  ->  ( ph  <->  x  =  z ) ) )
1211alimdv 1890 . . . . . 6  |-  ( z  =  y  ->  ( A. x ( ph  <->  x  =  y )  ->  A. x
( ph  <->  x  =  z
) ) )
1312com12 30 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  (
z  =  y  ->  A. x ( ph  <->  x  =  z ) ) )
147, 13impbid 129 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( A. x ( ph  <->  x  =  z )  <->  z  =  y ) )
1514alrimiv 1885 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  A. z
( A. x (
ph 
<->  x  =  z )  <-> 
z  =  y ) )
16 uniabio 5206 . . 3  |-  ( A. z ( A. x
( ph  <->  x  =  z
)  <->  z  =  y )  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  y )
1715, 16syl 14 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
z  |  A. x
( ph  <->  x  =  z
) }  =  y )
181, 17eqtrid 2234 1  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1362    = wceq 1364    e. wcel 2160   {cab 2175   _Vcvv 2752   U.cuni 3824   iotacio 5194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-sn 3613  df-pr 3614  df-uni 3825  df-iota 5196
This theorem is referenced by:  iotauni  5208  iota1  5210  euiotaex  5212  iota4  5215  iota5  5217
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