Theorem frins2

Description: Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	frins2.1	$$⊢R\mathrm{Fr}A$$
	frins2.2	$$⊢R\mathrm{Se}A$$
	frins2.3	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
	frins2.4	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
Assertion	frins2	$$⊢y\in A\to \phi$$

Proof

Step	Hyp	Ref	Expression
1		frins2.1	$$⊢R\mathrm{Fr}A$$
2		frins2.2	$$⊢R\mathrm{Se}A$$
3		frins2.3	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
4		frins2.4	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
5		nfv	$$⊢Ⅎy\psi$$
6	1 2 5 3 4	frins2f	$$⊢y\in A\to \phi$$