Theorem bj-spim0

Description: A universal specialization result in deduction form, proved from ax-1 -- ax-6 , where the only DV condition is on x , y and where x should be nonfree in the new proposition ch (and in the context ph ). (Contributed by BJ, 4-Apr-2026)

	Ref	Expression
Hypotheses	bj-spim0.nf0	|- ( ph -> A. x ph )
	bj-spim0.nf	|- ( ph -> ( E. x ch -> ch ) )
	bj-spim0.is	|- ( ( ph /\ x = y ) -> ( ps -> ch ) )
Assertion	bj-spim0	|- ( ph -> ( A. x ps -> ch ) )

Proof

Step	Hyp	Ref	Expression
1		bj-spim0.nf0	|- ( ph -> A. x ph )
2		bj-spim0.nf	|- ( ph -> ( E. x ch -> ch ) )
3		bj-spim0.is	|- ( ( ph /\ x = y ) -> ( ps -> ch ) )
4		ax6ev	|- E. x x = y
5	4	a1i	|- ( ph -> E. x x = y )
6	1 2 5 3	bj-spim	|- ( ph -> ( A. x ps -> ch ) )