Theorem frins

Description: Well-Founded Induction Schema. If a property passes from all elements less than y of a well-founded class A to y itself (induction hypothesis), then the property holds for all elements of A . (Contributed by Scott Fenton, 6-Feb-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	frins.1	$$⊢R\mathrm{Fr}A$$
	frins.2	$$⊢R\mathrm{Se}A$$
	frins.3	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\underset{\dot{}}{[}z/y\underset{\dot{}}{]}\phi \to \phi \right)$$
Assertion	frins	$$⊢y\in A\to \phi$$

Proof

Step	Hyp	Ref	Expression
1		frins.1	$$⊢R\mathrm{Fr}A$$
2		frins.2	$$⊢R\mathrm{Se}A$$
3		frins.3	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\underset{\dot{}}{[}z/y\underset{\dot{}}{]}\phi \to \phi \right)$$
4	3	frinsg	$$⊢\left(R\mathrm{Fr}A\wedge R\mathrm{Se}A\right)\to \forall y\in A\phi$$
5	1 2 4	mp2an	$$⊢\forall y\in A\phi$$
6	5	rspec	$$⊢y\in A\to \phi$$