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 e. A -> A. x y e. A )
Assertion	hblem	|- ( z e. A -> A. x z e. A )

Proof

Step	Hyp	Ref	Expression
1		hblem.1	|- ( y e. A -> A. x y e. A )
2	1	hbsbw	|- ( [ z / y ] y e. A -> A. x [ z / y ] y e. A )
3		clelsb1	|- ( [ z / y ] y e. A <-> z e. A )
4	3	albii	|- ( A. x [ z / y ] y e. A <-> A. x z e. A )
5	2 3 4	3imtr3i	|- ( z e. A -> A. x z e. A )