Theorem hbsb3

Description: If y is not free in ph , x is not free in [ y / x ] ph . Usage of this theorem is discouraged because it depends on ax-13 . Check out bj-hbsb3v for a weaker version requiring fewer axioms. (Contributed by NM, 14-May-1993) (New usage is discouraged.)

	Ref	Expression
Hypothesis	hbsb3.1	$$⊢\phi \to \forall y\phi$$
Assertion	hbsb3	$$⊢\left[y/x\right]\phi \to \forall x\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		hbsb3.1	$$⊢\phi \to \forall y\phi$$
2	1	sbimi	$$⊢\left[y/x\right]\phi \to \left[y/x\right]\forall y\phi$$
3		hbsb2a	$$⊢\left[y/x\right]\forall y\phi \to \forall x\left[y/x\right]\phi$$
4	2 3	syl	$$⊢\left[y/x\right]\phi \to \forall x\left[y/x\right]\phi$$