Theorem hbn1fw

Description: Weak version of ax-10 from which we can prove any ax-10 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017) (Proof shortened by Wolf Lammen, 28-Feb-2018)

	Ref	Expression
Hypotheses	hbn1fw.1	|- ( A. x ph -> A. y A. x ph )
	hbn1fw.2	|- ( -. ps -> A. x -. ps )
	hbn1fw.3	|- ( A. y ps -> A. x A. y ps )
	hbn1fw.4	|- ( -. ph -> A. y -. ph )
	hbn1fw.5	|- ( -. A. y ps -> A. x -. A. y ps )
	hbn1fw.6	|- ( x = y -> ( ph <-> ps ) )
Assertion	hbn1fw	|- ( -. A. x ph -> A. x -. A. x ph )

Proof

Step	Hyp	Ref	Expression
1		hbn1fw.1	|- ( A. x ph -> A. y A. x ph )
2		hbn1fw.2	|- ( -. ps -> A. x -. ps )
3		hbn1fw.3	|- ( A. y ps -> A. x A. y ps )
4		hbn1fw.4	|- ( -. ph -> A. y -. ph )
5		hbn1fw.5	|- ( -. A. y ps -> A. x -. A. y ps )
6		hbn1fw.6	|- ( x = y -> ( ph <-> ps ) )
7	1 2 3 4 6	cbvalw	|- ( A. x ph <-> A. y ps )
8	7	notbii	|- ( -. A. x ph <-> -. A. y ps )
9	8 5	hbxfrbi	|- ( -. A. x ph -> A. x -. A. x ph )