Theorem wfis2fgOLD

Description: Obsolete proof of wfis2fg as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 11-Feb-2011)

	Ref	Expression
Hypotheses	wfis2fgOLD.1	|- F/ y ps
	wfis2fgOLD.2	|- ( y = z -> ( ph <-> ps ) )
	wfis2fgOLD.3	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
Assertion	wfis2fgOLD	|- ( ( R We A /\ R Se A ) -> A. y e. A ph )

Proof

Step	Hyp	Ref	Expression
1		wfis2fgOLD.1	|- F/ y ps
2		wfis2fgOLD.2	|- ( y = z -> ( ph <-> ps ) )
3		wfis2fgOLD.3	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) )
4		sbsbc	|- ( [ z / y ] ph <-> [. z / y ]. ph )
5	1 2	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 3	syl5bi	|- ( y e. A -> ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) )
9	8	wfisg	|- ( ( R We A /\ R Se A ) -> A. y e. A ph )