Theorem frins3

Description: Well-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	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
	frins3.2	$$⊢y=B\to \left(\phi \leftrightarrow \chi \right)$$
	frins3.3	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
Assertion	frins3	$$⊢\left(\left(R\mathrm{Fr}A\wedge R\mathrm{Se}A\right)\wedge B\in A\right)\to \chi$$

Proof

Step	Hyp	Ref	Expression
1		frins3.1	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
2		frins3.2	$$⊢y=B\to \left(\phi \leftrightarrow \chi \right)$$
3		frins3.3	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
4	3 1	frins2	$$⊢\left(R\mathrm{Fr}A\wedge R\mathrm{Se}A\right)\to \forall y\in A\phi$$
5	2	rspcv	$$⊢B\in A\to \left(\forall y\in A\phi \to \chi \right)$$
6	4 5	mpan9	$$⊢\left(\left(R\mathrm{Fr}A\wedge R\mathrm{Se}A\right)\wedge B\in A\right)\to \chi$$