Theorem wfis2f

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

	Ref	Expression
Hypotheses	wfis2f.1	$$⊢R\mathrm{We}A$$
	wfis2f.2	$$⊢R\mathrm{Se}A$$
	wfis2f.3	$$⊢Ⅎy\psi$$
	wfis2f.4	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
	wfis2f.5	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
Assertion	wfis2f	$$⊢y\in A\to \phi$$

Proof

Step	Hyp	Ref	Expression
1		wfis2f.1	$$⊢R\mathrm{We}A$$
2		wfis2f.2	$$⊢R\mathrm{Se}A$$
3		wfis2f.3	$$⊢Ⅎy\psi$$
4		wfis2f.4	$$⊢y=z\to \left(\phi \leftrightarrow \psi \right)$$
5		wfis2f.5	$$⊢y\in A\to \left(\forall z\in Pred\left(R,A,y\right)\psi \to \phi \right)$$
6	3 4 5	wfis2fg	$$⊢\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\to \forall y\in A\phi$$
7	1 2 6	mp2an	$$⊢\forall y\in A\phi$$
8	7	rspec	$$⊢y\in A\to \phi$$