Theorem tfis3

Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003)

	Ref	Expression
Hypotheses	tfis3.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
	tfis3.2	$$⊢x=A\to \left(\phi \leftrightarrow \chi \right)$$
	tfis3.3	$$⊢x\in \mathrm{On}\to \left(\forall y\in x\psi \to \phi \right)$$
Assertion	tfis3	$$⊢A\in \mathrm{On}\to \chi$$

Proof

Step	Hyp	Ref	Expression
1		tfis3.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		tfis3.2	$$⊢x=A\to \left(\phi \leftrightarrow \chi \right)$$
3		tfis3.3	$$⊢x\in \mathrm{On}\to \left(\forall y\in x\psi \to \phi \right)$$
4	1 3	tfis2	$$⊢x\in \mathrm{On}\to \phi$$
5	2 4	vtoclga	$$⊢A\in \mathrm{On}\to \chi$$