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Theorem iotaval 4928
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 4918 . 2  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
2 vex 2613 . . . . . . 7  |-  y  e. 
_V
3 sbeqalb 2879 . . . . . . . 8  |-  ( y  e.  _V  ->  (
( A. x (
ph 
<->  x  =  y )  /\  A. x (
ph 
<->  x  =  z ) )  ->  y  =  z ) )
4 equcomi 1633 . . . . . . . 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 7 . . . . . 6  |-  ( ( A. x ( ph  <->  x  =  y )  /\  A. x ( ph  <->  x  =  z ) )  -> 
z  =  y )
76ex 113 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  ( A. x ( ph  <->  x  =  z )  ->  z  =  y ) )
8 equequ2 1641 . . . . . . . . . 10  |-  ( y  =  z  ->  (
x  =  y  <->  x  =  z ) )
98equcoms 1636 . . . . . . . . 9  |-  ( z  =  y  ->  (
x  =  y  <->  x  =  z ) )
109bibi2d 230 . . . . . . . 8  |-  ( z  =  y  ->  (
( ph  <->  x  =  y
)  <->  ( ph  <->  x  =  z ) ) )
1110biimpd 142 . . . . . . 7  |-  ( z  =  y  ->  (
( ph  <->  x  =  y
)  ->  ( ph  <->  x  =  z ) ) )
1211alimdv 1802 . . . . . 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 127 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( A. x ( ph  <->  x  =  z )  <->  z  =  y ) )
1514alrimiv 1797 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  A. z
( A. x (
ph 
<->  x  =  z )  <-> 
z  =  y ) )
16 uniabio 4927 . . 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, 17syl5eq 2127 1  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434   {cab 2069   _Vcvv 2610   U.cuni 3621   iotacio 4915
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  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-sn 3422  df-pr 3423  df-uni 3622  df-iota 4917
This theorem is referenced by:  iotauni  4929  iota1  4931  euiotaex  4933  iota4  4935  iota5  4937
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