Theorem hbae

Description: All variables are effectively bound in an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker hbaev when possible. (Contributed by NM, 13-May-1993) (Proof shortened by Wolf Lammen, 21-Apr-2018) (New usage is discouraged.)

	Ref	Expression
Assertion	hbae	$$⊢\forall xx=y\to \forall z\forall xx=y$$

Proof

Step	Hyp	Ref	Expression
1		sp	$$⊢\forall xx=y\to x=y$$
2		axc9	$$⊢\neg \forall zz=x\to \left(\neg \forall zz=y\to \left(x=y\to \forall zx=y\right)\right)$$
3	1 2	syl7	$$⊢\neg \forall zz=x\to \left(\neg \forall zz=y\to \left(\forall xx=y\to \forall zx=y\right)\right)$$
4		axc11r	$$⊢\forall zz=x\to \left(\forall xx=y\to \forall zx=y\right)$$
5		axc11	$$⊢\forall xx=y\to \left(\forall xx=y\to \forall yx=y\right)$$
6	5	pm2.43i	$$⊢\forall xx=y\to \forall yx=y$$
7		axc11r	$$⊢\forall zz=y\to \left(\forall yx=y\to \forall zx=y\right)$$
8	6 7	syl5	$$⊢\forall zz=y\to \left(\forall xx=y\to \forall zx=y\right)$$
9	3 4 8	pm2.61ii	$$⊢\forall xx=y\to \forall zx=y$$
10	9	axc4i	$$⊢\forall xx=y\to \forall x\forall zx=y$$
11		ax-11	$$⊢\forall x\forall zx=y\to \forall z\forall xx=y$$
12	10 11	syl	$$⊢\forall xx=y\to \forall z\forall xx=y$$