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Metamath Proof Explorer

Theorem frins2f

Description: Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2011) (Revised by Mario Carneiro, 11-Dec-2016)

Ref Expression
Hypotheses frins2f.1 
|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
frins2f.2 
|- F/ y ps
frins2f.3 
|- ( y = z -> ( ph <-> ps ) )
Assertion frins2f 
|- ( ( R Fr A /\ R Se A ) -> A. y e. A ph )

Proof

Step Hyp Ref Expression
1 frins2f.1 
 |-  ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
2 frins2f.2 
 |-  F/ y ps
3 frins2f.3 
 |-  ( y = z -> ( ph <-> ps ) )
4 sbsbc 
 |-  ( [ z / y ] ph <-> [. z / y ]. ph )
5 2 3 sbiev 
 |-  ( [ z / y ] ph <-> ps )
6 4 5 bitr3i 
 |-  ( [. z / y ]. ph <-> ps )
7 6 ralbii 
 |-  ( A. z e. Pred ( R , A , y ) [. z / y ]. ph <-> A. z e. Pred ( R , A , y ) ps )
8 7 1 biimtrid 
 |-  ( y e. A -> ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) )
9 8 frinsg 
 |-  ( ( R Fr A /\ R Se A ) -> A. y e. A ph )