Metamath Proof Explorer

Theorem pm11.57

Description: Theorem *11.57 in WhiteheadRussell p. 165. (Contributed by Andrew Salmon, 24-May-2011)

Ref Expression
Assertion pm11.57 
|- ( A. x ph <-> A. x A. y ( ph /\ [ y / x ] ph ) )

Proof

Step Hyp Ref Expression
1 nfv 
 |-  F/ y ph
2 1 nfal 
 |-  F/ y A. x ph
3 sp 
 |-  ( A. x ph -> ph )
4 stdpc4 
 |-  ( A. x ph -> [ y / x ] ph )
5 3 4 jca 
 |-  ( A. x ph -> ( ph /\ [ y / x ] ph ) )
6 2 5 alrimi 
 |-  ( A. x ph -> A. y ( ph /\ [ y / x ] ph ) )
7 6 axc4i 
 |-  ( A. x ph -> A. x A. y ( ph /\ [ y / x ] ph ) )
8 simpl 
 |-  ( ( ph /\ [ y / x ] ph ) -> ph )
9 8 sps 
 |-  ( A. y ( ph /\ [ y / x ] ph ) -> ph )
10 9 alimi 
 |-  ( A. x A. y ( ph /\ [ y / x ] ph ) -> A. x ph )
11 7 10 impbii 
 |-  ( A. x ph <-> A. x A. y ( ph /\ [ y / x ] ph ) )