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Theorem iotaval 5057
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 5047 . 2  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
2 vex 2660 . . . . . . 7  |-  y  e. 
_V
3 sbeqalb 2933 . . . . . . . 8  |-  ( y  e.  _V  ->  (
( A. x (
ph 
<->  x  =  y )  /\  A. x (
ph 
<->  x  =  z ) )  ->  y  =  z ) )
4 equcomi 1663 . . . . . . . 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 114 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  ( A. x ( ph  <->  x  =  z )  ->  z  =  y ) )
8 equequ2 1672 . . . . . . . . . 10  |-  ( y  =  z  ->  (
x  =  y  <->  x  =  z ) )
98equcoms 1667 . . . . . . . . 9  |-  ( z  =  y  ->  (
x  =  y  <->  x  =  z ) )
109bibi2d 231 . . . . . . . 8  |-  ( z  =  y  ->  (
( ph  <->  x  =  y
)  <->  ( ph  <->  x  =  z ) ) )
1110biimpd 143 . . . . . . 7  |-  ( z  =  y  ->  (
( ph  <->  x  =  y
)  ->  ( ph  <->  x  =  z ) ) )
1211alimdv 1833 . . . . . 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 128 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( A. x ( ph  <->  x  =  z )  <->  z  =  y ) )
1514alrimiv 1828 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  A. z
( A. x (
ph 
<->  x  =  z )  <-> 
z  =  y ) )
16 uniabio 5056 . . 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 2159 1  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312    = wceq 1314    e. wcel 1463   {cab 2101   _Vcvv 2657   U.cuni 3702   iotacio 5044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-v 2659  df-sbc 2879  df-un 3041  df-sn 3499  df-pr 3500  df-uni 3703  df-iota 5046
This theorem is referenced by:  iotauni  5058  iota1  5060  euiotaex  5062  iota4  5064  iota5  5066
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