Metamath Proof Explorer


Theorem sb5ALT

Description: Equivalence for substitution. Alternate proof of sb5 . This proof is sb5ALTVD automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion sb5ALT
|- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )

Proof

Step Hyp Ref Expression
1 equsb1
 |-  [ y / x ] x = y
2 sban
 |-  ( [ y / x ] ( x = y /\ ph ) <-> ( [ y / x ] x = y /\ [ y / x ] ph ) )
3 2 simplbi2com
 |-  ( [ y / x ] ph -> ( [ y / x ] x = y -> [ y / x ] ( x = y /\ ph ) ) )
4 1 3 mpi
 |-  ( [ y / x ] ph -> [ y / x ] ( x = y /\ ph ) )
5 spsbe
 |-  ( [ y / x ] ( x = y /\ ph ) -> E. x ( x = y /\ ph ) )
6 4 5 syl
 |-  ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
7 hbs1
 |-  ( [ y / x ] ph -> A. x [ y / x ] ph )
8 simpr
 |-  ( ( x = y /\ ph ) -> ph )
9 8 a1i
 |-  ( E. x ( x = y /\ ph ) -> ( ( x = y /\ ph ) -> ph ) )
10 simpl
 |-  ( ( x = y /\ ph ) -> x = y )
11 10 a1i
 |-  ( E. x ( x = y /\ ph ) -> ( ( x = y /\ ph ) -> x = y ) )
12 sbequ1
 |-  ( x = y -> ( ph -> [ y / x ] ph ) )
13 12 com12
 |-  ( ph -> ( x = y -> [ y / x ] ph ) )
14 9 11 13 syl6c
 |-  ( E. x ( x = y /\ ph ) -> ( ( x = y /\ ph ) -> [ y / x ] ph ) )
15 7 14 exlimexi
 |-  ( E. x ( x = y /\ ph ) -> [ y / x ] ph )
16 6 15 impbii
 |-  ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )