Theorem bnj157

Description: Well-founded induction restricted to a set ( A e. _V ). The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf . (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj157.1	|- ( ps <-> A. y e. A ( y R x -> [. y / x ]. ph ) )
	bnj157.2	|- A e. _V
	bnj157.3	|- R Fr A
Assertion	bnj157	|- ( A. x e. A ( ps -> ph ) -> A. x e. A ph )

Proof

Step	Hyp	Ref	Expression
1		bnj157.1	|- ( ps <-> A. y e. A ( y R x -> [. y / x ]. ph ) )
2		bnj157.2	|- A e. _V
3		bnj157.3	|- R Fr A
4	2 1	bnj110	|- ( ( R Fr A /\ A. x e. A ( ps -> ph ) ) -> A. x e. A ph )
5	3 4	mpan	|- ( A. x e. A ( ps -> ph ) -> A. x e. A ph )