Theorem frpoins2fg

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

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

Proof

Step	Hyp	Ref	Expression
1		frpoins2fg.1	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
2		frpoins2fg.2	$$⊢Ⅎy\psi$$
3		frpoins2fg.3	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
4		sbsbc	$$⊢\left[z/y\right]\phi \leftrightarrow \underset{\dot{}}{[}z/y\underset{\dot{}}{]}\phi$$
5	2 3	sbiev	$$⊢\left[z/y\right]\phi \leftrightarrow \psi$$
6	4 5	bitr3i	$$⊢\underset{\dot{}}{[}z/y\underset{\dot{}}{]}\phi \leftrightarrow \psi$$
7	6	ralbii	$$⊢\forall z\in Pred\left(R,A,y\right)\underset{\dot{}}{[}z/y\underset{\dot{}}{]}\phi \leftrightarrow \forall z\in Pred\left(R,A,y\right)\psi$$
8	1	adantl	$$⊢\left(\left(R\mathrm{Fr}A\wedge R\mathrm{Po}A\wedge R\mathrm{Se}A\right)\wedge y\in A\right)\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
9	7 8	syl5bi	$$⊢\left(\left(R\mathrm{Fr}A\wedge R\mathrm{Po}A\wedge R\mathrm{Se}A\right)\wedge y\in A\right)\to \left(\forall z\in Pred\left(R,A,y\right)\underset{\dot{}}{[}z/y\underset{\dot{}}{]}\phi \to \phi \right)$$
10	9	frpoinsg	$$⊢\left(R\mathrm{Fr}A\wedge R\mathrm{Po}A\wedge R\mathrm{Se}A\right)\to \forall y\in A\phi$$