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Theorem iotaval 6201
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
StepHypRef Expression
1 dfiota2 6191 . 2  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
2 vex 2743 . . . . . . 7  |-  y  e. 
_V
3 sbeqalb 2987 . . . . . . . 8  |-  ( y  e.  _V  ->  (
( A. x (
ph 
<->  x  =  y )  /\  A. x (
ph 
<->  x  =  z ) )  ->  y  =  z ) )
4 equcomi 1822 . . . . . . . 8  |-  ( y  =  z  ->  z  =  y )
53, 4syl6 31 . . . . . . 7  |-  ( y  e.  _V  ->  (
( A. x (
ph 
<->  x  =  y )  /\  A. x (
ph 
<->  x  =  z ) )  ->  z  =  y ) )
62, 5ax-mp 10 . . . . . 6  |-  ( ( A. x ( ph  <->  x  =  y )  /\  A. x ( ph  <->  x  =  z ) )  -> 
z  =  y )
76ex 425 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  ( A. x ( ph  <->  x  =  z )  ->  z  =  y ) )
8 equequ2 1830 . . . . . . . . . 10  |-  ( y  =  z  ->  (
x  =  y  <->  x  =  z ) )
98eqcoms 2259 . . . . . . . . 9  |-  ( z  =  y  ->  (
x  =  y  <->  x  =  z ) )
109bibi2d 311 . . . . . . . 8  |-  ( z  =  y  ->  (
( ph  <->  x  =  y
)  <->  ( ph  <->  x  =  z ) ) )
1110biimpd 200 . . . . . . 7  |-  ( z  =  y  ->  (
( ph  <->  x  =  y
)  ->  ( ph  <->  x  =  z ) ) )
1211alimdv 2018 . . . . . 6  |-  ( z  =  y  ->  ( A. x ( ph  <->  x  =  y )  ->  A. x
( ph  <->  x  =  z
) ) )
1312com12 29 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  (
z  =  y  ->  A. x ( ph  <->  x  =  z ) ) )
147, 13impbid 185 . . . 4  |-  ( A. x ( ph  <->  x  =  y )  ->  ( A. x ( ph  <->  x  =  z )  <->  z  =  y ) )
1514alrimiv 2013 . . 3  |-  ( A. x ( ph  <->  x  =  y )  ->  A. z
( A. x (
ph 
<->  x  =  z )  <-> 
z  =  y ) )
16 uniabio 6200 . . 3  |-  ( A. z ( A. x
( ph  <->  x  =  z
)  <->  z  =  y )  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  y )
1715, 16syl 17 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  U. {
z  |  A. x
( ph  <->  x  =  z
) }  =  y )
181, 17syl5eq 2300 1  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532    = wceq 1619    e. wcel 1621   {cab 2242   _Vcvv 2740   U.cuni 3768   iotacio 6188
This theorem is referenced by:  iotauni  6202  iota1  6204  iotaex  6207  iota4  6208  iota5  6210  iotain  26950  iotaexeu  26951  iotasbc  26952  iotaequ  26962  iotavalb  26963  pm14.24  26965  sbiota1  26967
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-rex 2521  df-v 2742  df-sbc 2936  df-un 3099  df-sn 3587  df-pr 3588  df-uni 3769  df-iota 6190
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