Theorem hba1wOLD 1975

Description: Obsolete proof of hba1w 1974 as of 10-Oct-2021. (Contributed by NM, 9-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
hba1wOLD	|- ( 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 hba1wOLD

Step	Hyp	Ref	Expression
1		hbn1w.1	. . . . . . 7 |- ( x = y -> ( ph <-> ps ) )
2	1	cbvalvw 1969	. . . . . 6 |- ( A. x ph <-> A. y ps )
3	2	a1i 11	. . . . 5 |- ( x = y -> ( A. x ph <-> A. y ps ) )
4	3	notbid 308	. . . 4 |- ( x = y -> ( -. A. x ph <-> -. A. y ps ) )
5	4	spw 1967	. . 3 |- ( A. x -. A. x ph -> -. A. x ph )
6	5	con2i 134	. 2 |- ( A. x ph -> -. A. x -. A. x ph )
7	4	hbn1w 1973	. 2 |- ( -. A. x -. A. x ph -> A. x -. A. x -. A. x ph )
8	1	hbn1w 1973	. . . 4 |- ( -. A. x ph -> A. x -. A. x ph )
9	8	con1i 144	. . 3 |- ( -. A. x -. A. x ph -> A. x ph )
10	9	alimi 1739	. 2 |- ( A. x -. A. x -. A. x ph -> A. x A. x ph )
11	6, 7, 10	3syl 18	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: (None)