Theorem frpoins3g

Description: Founded Partial Induction schema, using implicit substitution. (Contributed by Scott Fenton, 19-Aug-2024)

	Ref	Expression
Hypotheses	frpoins3g.1	$$⊢x\in A\to \left(\forall y\in Pred\left(R,A,x\right)\psi \to \phi \right)$$
	frpoins3g.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
	frpoins3g.3	$$⊢x=B\to \left(\phi \leftrightarrow \chi \right)$$
Assertion	frpoins3g	$$⊢\left(\left(R\mathrm{Fr}A\wedge R\mathrm{Po}A\wedge R\mathrm{Se}A\right)\wedge B\in A\right)\to \chi$$

Proof

Step	Hyp	Ref	Expression
1		frpoins3g.1	$$⊢x\in A\to \left(\forall y\in Pred\left(R,A,x\right)\psi \to \phi \right)$$
2		frpoins3g.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3		frpoins3g.3	$$⊢x=B\to \left(\phi \leftrightarrow \chi \right)$$
4	1 2	frpoins2g	$$⊢\left(R\mathrm{Fr}A\wedge R\mathrm{Po}A\wedge R\mathrm{Se}A\right)\to \forall x\in A\phi$$
5	3	rspccva	$$⊢\left(\forall x\in A\phi \wedge B\in A\right)\to \chi$$
6	4 5	sylan	$$⊢\left(\left(R\mathrm{Fr}A\wedge R\mathrm{Po}A\wedge R\mathrm{Se}A\right)\wedge B\in A\right)\to \chi$$