Theorem frins2fg

Description: Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2011) (Revised by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Hypotheses	frins2fg.1	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
	frins2fg.2	$$⊢Ⅎy\psi$$
	frins2fg.3	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	frins2fg	$$⊢\left(R\mathrm{Fr}A\wedge R\mathrm{Se}A\right)\to \forall y\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		frins2fg.1	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
2		frins2fg.2	$$⊢Ⅎy\psi$$
3		frins2fg.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	7 1	syl5bi	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\underset{\dot{}}{[}z/y\underset{\dot{}}{]}\phi \to \phi \right)$$
9	8	frinsg	$$⊢\left(R\mathrm{Fr}A\wedge R\mathrm{Se}A\right)\to \forall y\in A\phi$$