Theorem hbab

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995) Add disjoint variable condition to avoid ax-13 . See hbabg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

	Ref	Expression
Hypothesis	hbab.1	|- ( ph -> A. x ph )
Assertion	hbab	|- ( z e. { y | ph } -> A. x z e. { y | ph } )

Proof

Step	Hyp	Ref	Expression
1		hbab.1	|- ( ph -> A. x ph )
2		df-clab	|- ( z e. { y | ph } <-> [ z / y ] ph )
3	1	hbsbw	|- ( [ z / y ] ph -> A. x [ z / y ] ph )
4	2 3	hbxfrbi	|- ( z e. { y | ph } -> A. x z e. { y | ph } )