Theorem nfaba1

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016) Add disjoint variable condition to avoid ax-13 . See nfaba1g for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024) Avoid ax-6 , ax-7 , ax-12 . (Revised by SN, 14-May-2025)

	Ref	Expression
Assertion	nfaba1	|- F/_ x { y | A. x ph }

Proof

Step	Hyp	Ref	Expression
1		df-clab	|- ( z e. { y | A. x ph } <-> [ z / y ] A. x ph )
2		sbal	|- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )
3		nfa1	|- F/ x A. x [ z / y ] ph
4	2 3	nfxfr	|- F/ x [ z / y ] A. x ph
5	1 4	nfxfr	|- F/ x z e. { y | A. x ph }
6	5	nfci	|- F/_ x { y | A. x ph }