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Metamath Proof Explorer

Theorem iota4

Description: Theorem *14.22 in WhiteheadRussell p. 190. (Contributed by Andrew Salmon, 12-Jul-2011)

Ref Expression
Assertion iota4 
|- ( E! x ph -> [. ( iota x ph ) / x ]. ph )

Proof

Step Hyp Ref Expression
1 eu6 
 |-  ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2 biimpr 
 |-  ( ( ph <-> x = z ) -> ( x = z -> ph ) )
3 2 alimi 
 |-  ( A. x ( ph <-> x = z ) -> A. x ( x = z -> ph ) )
4 sb6 
 |-  ( [ z / x ] ph <-> A. x ( x = z -> ph ) )
5 3 4 sylibr 
 |-  ( A. x ( ph <-> x = z ) -> [ z / x ] ph )
6 iotaval 
 |-  ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
7 6 eqcomd 
 |-  ( A. x ( ph <-> x = z ) -> z = ( iota x ph ) )
8 dfsbcq2 
 |-  ( z = ( iota x ph ) -> ( [ z / x ] ph <-> [. ( iota x ph ) / x ]. ph ) )
9 7 8 syl 
 |-  ( A. x ( ph <-> x = z ) -> ( [ z / x ] ph <-> [. ( iota x ph ) / x ]. ph ) )
10 5 9 mpbid 
 |-  ( A. x ( ph <-> x = z ) -> [. ( iota x ph ) / x ]. ph )
11 10 exlimiv 
 |-  ( E. z A. x ( ph <-> x = z ) -> [. ( iota x ph ) / x ]. ph )
12 1 11 sylbi 
 |-  ( E! x ph -> [. ( iota x ph ) / x ]. ph )