Theorem hbn1fw 1972

Description: Weak version of ax-10 2019 from which we can prove any ax-10 2019 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.)

Hypotheses

Ref	Expression
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

Ref	Expression
hbn1fw	|- ( -. A. x ph -> A. x -. A. x ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem hbn1fw

Step	Hyp	Ref	Expression
1		hbn1fw.1	. . . 4 |- ( A. x ph -> A. y A. x ph )
2		hbn1fw.2	. . . 4 |- ( -. ps -> A. x -. ps )
3		hbn1fw.3	. . . 4 |- ( A. y ps -> A. x A. y ps )
4		hbn1fw.4	. . . 4 |- ( -. ph -> A. y -. ph )
5		hbn1fw.6	. . . 4 |- ( x = y -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvalw 1968	. . 3 |- ( A. x ph <-> A. y ps )
7	6	notbii 310	. 2 |- ( -. A. x ph <-> -. A. y ps )
8		hbn1fw.5	. 2 |- ( -. A. y ps -> A. x -. A. y ps )
9	7, 8	hbxfrbi 1752	1 |- ( -. A. x ph -> A. x -. A. x ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  hbn1w  1973