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Metamath Proof Explorer

Theorem iota4an

Description: Theorem *14.23 in WhiteheadRussell p. 191. (Contributed by Andrew Salmon, 12-Jul-2011)

Ref Expression
Assertion iota4an 
|- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )

Proof

Step Hyp Ref Expression
1 iota4 
 |-  ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) )
2 iotaex 
 |-  ( iota x ( ph /\ ps ) ) e. _V
3 simpl 
 |-  ( ( ph /\ ps ) -> ph )
4 3 sbcth 
 |-  ( ( iota x ( ph /\ ps ) ) e. _V -> [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) )
5 2 4 ax-mp 
 |-  [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph )
6 sbcimg 
 |-  ( ( iota x ( ph /\ ps ) ) e. _V -> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) <-> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) ) )
7 2 6 ax-mp 
 |-  ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ( ph /\ ps ) -> ph ) <-> ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph ) )
8 5 7 mpbi 
 |-  ( [. ( iota x ( ph /\ ps ) ) / x ]. ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )
9 1 8 syl 
 |-  ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )