Theorem hbsb2

Description: Bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 14-May-1993) (New usage is discouraged.)

	Ref	Expression
Assertion	hbsb2	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \to \forall x\left[y/x\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sb4b	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$
2		sb2	$$⊢\forall x\left(x=y\to \phi \right)\to \left[y/x\right]\phi$$
3	2	axc4i	$$⊢\forall x\left(x=y\to \phi \right)\to \forall x\left[y/x\right]\phi$$
4	1 3	biimtrdi	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \to \forall x\left[y/x\right]\phi \right)$$