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Theorem iotavalb 27630
Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5230. (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 5230 . 2  |-  ( A. x ( ph  <->  x  =  y )  ->  ( iota x ph )  =  y )
2 iotasbc 27619 . . . 4  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  z ]. z  =  y  <->  E. z ( A. x ( ph  <->  x  =  z )  /\  z  =  y ) ) )
3 iotaexeu 27618 . . . . 5  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
4 eqsbc3 3030 . . . . 5  |-  ( ( iota x ph )  e.  _V  ->  ( [. ( iota x ph )  /  z ]. z  =  y  <->  ( iota x ph )  =  y
) )
53, 4syl 15 . . . 4  |-  ( E! x ph  ->  ( [. ( iota x ph )  /  z ]. z  =  y  <->  ( iota x ph )  =  y
) )
62, 5bitr3d 246 . . 3  |-  ( E! x ph  ->  ( E. z ( A. x
( ph  <->  x  =  z
)  /\  z  =  y )  <->  ( iota x ph )  =  y ) )
7 equequ2 1649 . . . . . . 7  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
87bibi2d 309 . . . . . 6  |-  ( z  =  y  ->  (
( ph  <->  x  =  z
)  <->  ( ph  <->  x  =  y ) ) )
98albidv 1611 . . . . 5  |-  ( z  =  y  ->  ( A. x ( ph  <->  x  =  z )  <->  A. x
( ph  <->  x  =  y
) ) )
109biimpac 472 . . . 4  |-  ( ( A. x ( ph  <->  x  =  z )  /\  z  =  y )  ->  A. x ( ph  <->  x  =  y ) )
1110exlimiv 1666 . . 3  |-  ( E. z ( A. x
( ph  <->  x  =  z
)  /\  z  =  y )  ->  A. x
( ph  <->  x  =  y
) )
126, 11syl6bir 220 . 2  |-  ( E! x ph  ->  (
( iota x ph )  =  y  ->  A. x
( ph  <->  x  =  y
) ) )
131, 12impbid2 195 1  |-  ( E! x ph  ->  ( 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 1527   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   _Vcvv 2788   [.wsbc 2991   iotacio 5217
This theorem is referenced by:  iotavalsb  27633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-sbc 2992  df-un 3157  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219
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