Theorem wfis3

Description: Well-Ordered Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011)

	Ref	Expression
Hypotheses	wfis3.1	$$⊢R\mathrm{We}A$$
	wfis3.2	$$⊢R\mathrm{Se}A$$
	wfis3.3	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
	wfis3.4	$$⊢y=B\to \left(\phi \leftrightarrow \chi \right)$$
	wfis3.5	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
Assertion	wfis3	$$⊢B\in A\to \chi$$

Proof

Step	Hyp	Ref	Expression
1		wfis3.1	$$⊢R\mathrm{We}A$$
2		wfis3.2	$$⊢R\mathrm{Se}A$$
3		wfis3.3	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
4		wfis3.4	$$⊢y=B\to \left(\phi \leftrightarrow \chi \right)$$
5		wfis3.5	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
6	1 2 3 5	wfis2	$$⊢y\in A\to \phi$$
7	4 6	vtoclga	$$⊢B\in A\to \chi$$