Theorem frins3

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

	Ref	Expression
Hypotheses	frins3.1	$$⊢R\mathrm{Fr}A$$
	frins3.2	$$⊢R\mathrm{Se}A$$
	frins3.3	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
	frins3.4	$$⊢y=B\to \left(\phi \leftrightarrow \chi \right)$$
	frins3.5	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
Assertion	frins3	$$⊢B\in A\to \chi$$

Proof

Step	Hyp	Ref	Expression
1		frins3.1	$$⊢R\mathrm{Fr}A$$
2		frins3.2	$$⊢R\mathrm{Se}A$$
3		frins3.3	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
4		frins3.4	$$⊢y=B\to \left(\phi \leftrightarrow \chi \right)$$
5		frins3.5	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
6	1 2 3 5	frins2	$$⊢y\in A\to \phi$$
7	4 6	vtoclga	$$⊢B\in A\to \chi$$