Theorem wfis2

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

	Ref	Expression
Hypotheses	wfis2.1	$$⊢R\mathrm{We}A$$
	wfis2.2	$$⊢R\mathrm{Se}A$$
	wfis2.3	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
	wfis2.4	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
Assertion	wfis2	$$⊢y\in A\to \phi$$

Proof

Step	Hyp	Ref	Expression
1		wfis2.1	$$⊢R\mathrm{We}A$$
2		wfis2.2	$$⊢R\mathrm{Se}A$$
3		wfis2.3	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
4		wfis2.4	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
5	3 4	wfis2g	$$⊢\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\to \forall y\in A\phi$$
6	1 2 5	mp2an	$$⊢\forall y\in A\phi$$
7	6	rspec	$$⊢y\in A\to \phi$$