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Theorem iotavalb 27030
Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6264. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotavalb  |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y )  <->  ( iota x ph )  =  y ) )
Distinct variable group:    x, y
Dummy variable  z is distinct from all other variables.
Allowed substitution hints:    ph( x, y)

Proof of Theorem iotavalb
StepHypRef Expression
1 iotaval 6264 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
2 iotasbc 27019 . . . 4  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  z ]. z  =  y  <->  E. z ( A. x ( ph  <->  x  =  z )  /\  z  =  y ) ) )
3 iotaexeu 27018 . . . . 5  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
4 eqsbc3 3032 . . . . 5  |-  ( ( iota x ph )  e.  _V  ->  ( [. ( iota x ph )  /  z ]. z  =  y  <->  ( iota x ph )  =  y
) )
53, 4syl 17 . . . 4  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  z ]. z  =  y  <->  ( iota x ph )  =  y
) )
62, 5bitr3d 248 . . 3  |-  ( E! x ph  ->  ( E. z ( A. x
( ph  <->  x  =  z
)  /\  z  =  y )  <->  ( iota x ph )  =  y ) )
7 equequ2 1650 . . . . . . 7  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
87bibi2d 311 . . . . . 6  |-  ( z  =  y  ->  (
( ph  <->  x  =  z
)  <->  ( ph  <->  x  =  y ) ) )
98albidv 1612 . . . . 5  |-  ( z  =  y  ->  ( A. x ( ph  <->  x  =  z )  <->  A. x
( ph  <->  x  =  y
) ) )
109biimpac 474 . . . 4  |-  ( ( A. x ( ph  <->  x  =  z )  /\  z  =  y )  ->  A. x ( ph  <->  x  =  y ) )
1110exlimiv 1667 . . 3  |-  ( E. z ( A. x
( ph  <->  x  =  z
)  /\  z  =  y )  ->  A. x
( ph  <->  x  =  y
) )
126, 11syl6bir 222 . 2  |-  ( E! x ph  ->  (
( iota x ph )  =  y  ->  A. x
( ph  <->  x  =  y
) ) )
131, 12impbid2 197 1  |-  ( E! x ph  ->  ( 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 1528   E.wex 1529    = wceq 1624    e. wcel 1685   E!weu 2145   _Vcvv 2790   [.wsbc 2993   iotacio 6251
This theorem is referenced by:  iotavalsb  27033
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  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 6253
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