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Metamath Proof Explorer

Theorem wfis2fg

Description: Well-Ordered Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011) (Proof shortened by Scott Fenton, 17-Nov-2024)

Ref Expression
Hypotheses wfis2fg.1 
|- F/ y ps
wfis2fg.2 
|- ( y = z -> ( ph <-> ps ) )
wfis2fg.3 
|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
Assertion wfis2fg 
|- ( ( R We A /\ R Se A ) -> A. y e. A ph )

Proof

Step Hyp Ref Expression
1 wfis2fg.1 
 |-  F/ y ps
2 wfis2fg.2 
 |-  ( y = z -> ( ph <-> ps ) )
3 wfis2fg.3 
 |-  ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
4 wefr 
 |-  ( R We A -> R Fr A )
5 4 adantr 
 |-  ( ( R We A /\ R Se A ) -> R Fr A )
6 weso 
 |-  ( R We A -> R Or A )
7 sopo 
 |-  ( R Or A -> R Po A )
8 6 7 syl 
 |-  ( R We A -> R Po A )
9 8 adantr 
 |-  ( ( R We A /\ R Se A ) -> R Po A )
10 simpr 
 |-  ( ( R We A /\ R Se A ) -> R Se A )
11 3 1 2 frpoins2fg 
 |-  ( ( R Fr A /\ R Po A /\ R Se A ) -> A. y e. A ph )
12 5 9 10 11 syl3anc 
 |-  ( ( R We A /\ R Se A ) -> A. y e. A ph )