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Metamath Proof Explorer

Theorem iotavalb

Description: Theorem *14.202 in WhiteheadRussell p. 189. A biconditional version of iotaval . (Contributed by Andrew Salmon, 11-Jul-2011)

Ref Expression
Assertion iotavalb 
|- ( E! x ph -> ( A. x ( ph <-> x = y ) <-> ( iota x ph ) = y ) )

Proof

Step Hyp Ref Expression
1 iotaval 
 |-  ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
2 iotasbc 
 |-  ( E! x ph -> ( [. ( iota x ph ) / z ]. z = y <-> E. z ( A. x ( ph <-> x = z ) /\ z = y ) ) )
3 iotaexeu 
 |-  ( E! x ph -> ( iota x ph ) e. _V )
4 eqsbc1 
 |-  ( ( iota x ph ) e. _V -> ( [. ( iota x ph ) / z ]. z = y <-> ( iota x ph ) = y ) )
5 3 4 syl 
 |-  ( E! x ph -> ( [. ( iota x ph ) / z ]. z = y <-> ( iota x ph ) = y ) )
6 2 5 bitr3d 
 |-  ( E! x ph -> ( E. z ( A. x ( ph <-> x = z ) /\ z = y ) <-> ( iota x ph ) = y ) )
7 equequ2 
 |-  ( z = y -> ( x = z <-> x = y ) )
8 7 bibi2d 
 |-  ( z = y -> ( ( ph <-> x = z ) <-> ( ph <-> x = y ) ) )
9 8 albidv 
 |-  ( z = y -> ( A. x ( ph <-> x = z ) <-> A. x ( ph <-> x = y ) ) )
10 9 biimpac 
 |-  ( ( A. x ( ph <-> x = z ) /\ z = y ) -> A. x ( ph <-> x = y ) )
11 10 exlimiv 
 |-  ( E. z ( A. x ( ph <-> x = z ) /\ z = y ) -> A. x ( ph <-> x = y ) )
12 6 11 biimtrrdi 
 |-  ( E! x ph -> ( ( iota x ph ) = y -> A. x ( ph <-> x = y ) ) )
13 1 12 impbid2 
 |-  ( E! x ph -> ( A. x ( ph <-> x = y ) <-> ( iota x ph ) = y ) )