Theorem hblem

Description: Change the free variable of a hypothesis builder. (Contributed by NM, 21-Jun-1993) (Revised by Andrew Salmon, 11-Jul-2011) Add disjoint variable condition to avoid ax-13 . See hblemg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

	Ref	Expression
Hypothesis	hblem.1	$$⊢y\in A\to \forall xy\in A$$
Assertion	hblem	$$⊢z\in A\to \forall xz\in A$$

Proof

Step	Hyp	Ref	Expression
1		hblem.1	$$⊢y\in A\to \forall xy\in A$$
2	1	hbsbw	$$⊢\left[z/y\right]y\in A\to \forall x\left[z/y\right]y\in A$$
3		clelsb1	$$⊢\left[z/y\right]y\in A\leftrightarrow z\in A$$
4	3	albii	$$⊢\forall x\left[z/y\right]y\in A\leftrightarrow \forall xz\in A$$
5	2 3 4	3imtr3i	$$⊢z\in A\to \forall xz\in A$$