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Theorem iotavalb 29689
Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5397. (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 5397 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
2 iotasbc 29678 . . . 4  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  z ]. z  =  y  <->  E. z ( A. x ( ph  <->  x  =  z )  /\  z  =  y ) ) )
3 iotaexeu 29677 . . . . 5  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
4 eqsbc3 3231 . . . . 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 255 . . 3  |-  ( E! x ph  ->  ( E. z ( A. x
( ph  <->  x  =  z
)  /\  z  =  y )  <->  ( iota x ph )  =  y ) )
7 equequ2 1737 . . . . . . 7  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
87bibi2d 318 . . . . . 6  |-  ( z  =  y  ->  (
( ph  <->  x  =  z
)  <->  ( ph  <->  x  =  y ) ) )
98albidv 1679 . . . . 5  |-  ( z  =  y  ->  ( A. x ( ph  <->  x  =  z )  <->  A. x
( ph  <->  x  =  y
) ) )
109biimpac 486 . . . 4  |-  ( ( A. x ( ph  <->  x  =  z )  /\  z  =  y )  ->  A. x ( ph  <->  x  =  y ) )
1110exlimiv 1688 . . 3  |-  ( E. z ( A. x
( ph  <->  x  =  z
)  /\  z  =  y )  ->  A. x
( ph  <->  x  =  y
) )
126, 11syl6bir 229 . 2  |-  ( E! x ph  ->  (
( iota x ph )  =  y  ->  A. x
( ph  <->  x  =  y
) ) )
131, 12impbid2 204 1  |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y )  <->  ( iota x ph )  =  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   E!weu 2253   _Vcvv 2977   [.wsbc 3191   iotacio 5384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rex 2726  df-v 2979  df-sbc 3192  df-un 3338  df-sn 3883  df-pr 3885  df-uni 4097  df-iota 5386
This theorem is referenced by:  iotavalsb  29692
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