Theorem frpoins2g

Description: Founded Partial Induction schema, using implicit substitution. (Contributed by Scott Fenton, 24-Aug-2022)

	Ref	Expression
Hypotheses	frpoins2g.1	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
	frpoins2g.3	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	frpoins2g	$$⊢\left(R\mathrm{Fr}A\wedge R\mathrm{Po}A\wedge R\mathrm{Se}A\right)\to \forall y\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		frpoins2g.1	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
2		frpoins2g.3	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
3		nfv	$$⊢Ⅎy\psi$$
4	1 3 2	frpoins2fg	$$⊢\left(R\mathrm{Fr}A\wedge R\mathrm{Po}A\wedge R\mathrm{Se}A\right)\to \forall y\in A\phi$$