Theorem spfw 1965

Description: Weak version of sp 2053. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)

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
spfw	|- ( A. x ph -> ph )

Distinct variable group:   x, y

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

Proof of Theorem spfw

Step	Hyp	Ref	Expression
1		spfw.2	. . 3 |- ( A. x ph -> A. y A. x ph )
2		spfw.1	. . 3 |- ( -. ps -> A. x -. ps )
3		spfw.4	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	biimpd 219	. . 3 |- ( x = y -> ( ph -> ps ) )
5	1, 2, 4	cbvaliw 1933	. 2 |- ( A. x ph -> A. y ps )
6		spfw.3	. . 3 |- ( -. ph -> A. y -. ph )
7	3	biimprd 238	. . . 4 |- ( x = y -> ( ps -> ph ) )
8	7	equcoms 1947	. . 3 |- ( y = x -> ( ps -> ph ) )
9	6, 8	spimw 1926	. 2 |- ( A. y ps -> ph )
10	5, 9	syl 17	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:  spw  1967