Theorem spfw

Description: Weak version of sp . 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)

	Ref	Expression
Hypotheses	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	spfw	|- ( A. x ph -> ph )

Proof

Step	Hyp	Ref	Expression
1		spfw.1	|- ( -. ps -> A. x -. ps )
2		spfw.2	|- ( A. x ph -> A. y A. x ph )
3		spfw.3	|- ( -. ph -> A. y -. ph )
4		spfw.4	|- ( x = y -> ( ph <-> ps ) )
5	4	biimpd	|- ( x = y -> ( ph -> ps ) )
6	2 1 5	cbvaliw	|- ( A. x ph -> A. y ps )
7	4	biimprd	|- ( x = y -> ( ps -> ph ) )
8	7	equcoms	|- ( y = x -> ( ps -> ph ) )
9	3 8	spimw	|- ( A. y ps -> ph )
10	6 9	syl	|- ( A. x ph -> ph )