Theorem spfwOLD 1966

Description: Obsolete proof of spfw 1965 as of 10-Oct-2021. (Contributed by NM, 19-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
spfw.1	|- ( -. ps -> A. x -. ps )
spfw.2	|- ( A. x ph -> A. y A. x ph )
spfw.3	|- ( -. ph -> A. y -. ph )
spfw.4	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem spfwOLD

Step	Hyp	Ref	Expression
1		spfw.2	. . 3 |- ( A. x ph -> A. y A. x ph )
2		alim 1738	. . 3 |- ( A. y ( A. x ph -> ps ) -> ( A. y A. x ph -> A. y ps ) )
3		spfw.3	. . . 4 |- ( -. ph -> A. y -. ph )
4		spfw.4	. . . . . 6 |- ( x = y -> ( ph <-> ps ) )
5	4	biimprd 238	. . . . 5 |- ( x = y -> ( ps -> ph ) )
6	5	equcoms 1947	. . . 4 |- ( y = x -> ( ps -> ph ) )
7	3, 6	spimw 1926	. . 3 |- ( A. y ps -> ph )
8	1, 2, 7	syl56 36	. 2 |- ( A. y ( A. x ph -> ps ) -> ( A. x ph -> ph ) )
9		spfw.1	. . 3 |- ( -. ps -> A. x -. ps )
10	4	biimpd 219	. . 3 |- ( x = y -> ( ph -> ps ) )
11	9, 10	spimw 1926	. 2 |- ( A. x ph -> ps )
12	8, 11	mpg 1724	1 |- ( A. x ph -> 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)