Theorem hbn1w 1973

Description: Weak version of hbn1 2020. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Hypothesis

Ref	Expression
hbn1w.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
hbn1w	|- ( -. A. x ph -> A. x -. A. x ph )

Distinct variable groups:   ph, y   ps, x   x, y

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

Proof of Theorem hbn1w

Step	Hyp	Ref	Expression
1		ax-5 1839	. 2 |- ( A. x ph -> A. y A. x ph )
2		ax-5 1839	. 2 |- ( -. ps -> A. x -. ps )
3		ax-5 1839	. 2 |- ( A. y ps -> A. x A. y ps )
4		ax-5 1839	. 2 |- ( -. ph -> A. y -. ph )
5		ax-5 1839	. 2 |- ( -. A. y ps -> A. x -. A. y ps )
6		hbn1w.1	. 2 |- ( x = y -> ( ph <-> ps ) )
7	1, 2, 3, 4, 5, 6	hbn1fw 1972	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:  hba1w  1974  hba1wOLD  1975  hbe1w  1976  ax10w  2006  wl-naevhba1v  33304