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Theorem iotaval 5232
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 5222 . 2  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
2 vex 2793 . . . . . . 7  |-  y  e. 
_V
3 sbeqalb 3045 . . . . . . . 8  |-  ( y  e.  _V  ->  (
( A. x (
ph 
<->  x  =  y )  /\  A. x (
ph 
<->  x  =  z ) )  ->  y  =  z ) )
4 equcomi 1648 . . . . . . . 8  |-  ( y  =  z  ->  z  =  y )
53, 4syl6 29 . . . . . . 7  |-  ( y  e.  _V  ->  (
( A. x (
ph 
<->  x  =  y )  /\  A. x (
ph 
<->  x  =  z ) )  ->  z  =  y ) )
62, 5ax-mp 8 . . . . . 6  |-  ( ( A. x ( ph  <->  x  =  y )  /\  A. x ( ph  <->  x  =  z ) )  -> 
z  =  y )
76ex 423 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  ( A. x ( ph  <->  x  =  z )  ->  z  =  y ) )
8 equequ2 1651 . . . . . . . . . 10  |-  ( y  =  z  ->  (
x  =  y  <->  x  =  z ) )
98eqcoms 2288 . . . . . . . . 9  |-  ( z  =  y  ->  (
x  =  y  <->  x  =  z ) )
109bibi2d 309 . . . . . . . 8  |-  ( z  =  y  ->  (
( ph  <->  x  =  y
)  <->  ( ph  <->  x  =  z ) ) )
1110biimpd 198 . . . . . . 7  |-  ( z  =  y  ->  (
( ph  <->  x  =  y
)  ->  ( ph  <->  x  =  z ) ) )
1211alimdv 1609 . . . . . 6  |-  ( z  =  y  ->  ( A. x ( ph  <->  x  =  y )  ->  A. x
( ph  <->  x  =  z
) ) )
1312com12 27 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  (
z  =  y  ->  A. x ( ph  <->  x  =  z ) ) )
147, 13impbid 183 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( A. x ( ph  <->  x  =  z )  <->  z  =  y ) )
1514alrimiv 1619 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  A. z
( A. x (
ph 
<->  x  =  z )  <-> 
z  =  y ) )
16 uniabio 5231 . . 3  |-  ( A. z ( A. x
( ph  <->  x  =  z
)  <->  z  =  y )  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  y )
1715, 16syl 15 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
z  |  A. x
( ph  <->  x  =  z
) }  =  y )
181, 17syl5eq 2329 1  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1529    = wceq 1625    e. wcel 1686   {cab 2271   _Vcvv 2790   U.cuni 3829   iotacio 5219
This theorem is referenced by:  iotauni  5233  iota1  5235  iotaex  5238  iota4  5239  iota5  5241  iotain  27628  iotaexeu  27629  iotasbc  27630  iotaequ  27640  iotavalb  27641  pm14.24  27643  sbiota1  27645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-rex 2551  df-v 2792  df-sbc 2994  df-un 3159  df-sn 3648  df-pr 3649  df-uni 3830  df-iota 5221
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