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Theorem iotavalb 27645
Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5458. (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
Allowed substitution hints:    ph( x, y)

Proof of Theorem iotavalb
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 iotaval 5458 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
2 iotasbc 27634 . . . 4  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  z ]. z  =  y  <->  E. z ( A. x ( ph  <->  x  =  z )  /\  z  =  y ) ) )
3 iotaexeu 27633 . . . . 5  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
4 eqsbc3 3206 . . . . 5  |-  ( ( iota x ph )  e.  _V  ->  ( [. ( iota x ph )  /  z ]. z  =  y  <->  ( iota x ph )  =  y
) )
53, 4syl 16 . . . 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 1700 . . . . . . 7  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
87bibi2d 311 . . . . . 6  |-  ( z  =  y  ->  (
( ph  <->  x  =  z
)  <->  ( ph  <->  x  =  y ) ) )
98albidv 1636 . . . . 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 1645 . . 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 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1727   E!weu 2287   _Vcvv 2962   [.wsbc 3167   iotacio 5445
This theorem is referenced by:  iotavalsb  27648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rex 2717  df-v 2964  df-sbc 3168  df-un 3311  df-sn 3844  df-pr 3845  df-uni 4040  df-iota 5447
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