Theorem spfw 1691

Description: Weak version of sp 1747. 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. TO DO: Do we need this theorem? If not, maybe it should be deleted. (Contributed by NM, 19-Apr-2017.)

Hypotheses

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

Assertion

Ref	Expression
spfw	|- (A.xph -> 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.xph -> A.yA.xph)
2		ax-5 1557	. . 3 |- (A.y(A.xph -> ps) -> (A.yA.xph -> A.yps))
3		spfw.3	. . . 4 |- (-. ph -> A.y -. ph)
4		spfw.4	. . . . . 6 |- (x = y -> (ph <-> ps))
5	4	biimprd 214	. . . . 5 |- (x = y -> (ps -> ph))
6	5	equcoms 1681	. . . 4 |- (y = x -> (ps -> ph))
7	3, 6	spimw 1668	. . 3 |- (A.yps -> ph)
8	1, 2, 7	syl56 30	. 2 |- (A.y(A.xph -> ps) -> (A.xph -> ph))
9		spfw.1	. . 3 |- (-. ps -> A.x -. ps)
10	4	biimpd 198	. . 3 |- (x = y -> (ph -> ps))
11	9, 10	spimw 1668	. 2 |- (A.xph -> ps)
12	8, 11	mpg 1548	1 |- (A.xph -> ph)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  spnfwOLD  1692