Theorem bj-wnf1

Description: When ph is substituted for ps , this is the first half of nonfreness ( . -> A. ) of the weak form of nonfreeness ( E. -> A. ) . (Contributed by BJ, 9-Dec-2023)

	Ref	Expression
Assertion	bj-wnf1	|- ( ( E. x ph -> A. x ps ) -> A. x ( E. x ph -> A. x ps ) )

Proof

Step	Hyp	Ref	Expression
1		bj-modal4e	|- ( E. x E. x ph -> E. x ph )
2		hba1	|- ( A. x ps -> A. x A. x ps )
3	1 2	imim12i	|- ( ( E. x ph -> A. x ps ) -> ( E. x E. x ph -> A. x A. x ps ) )
4		19.38	|- ( ( E. x E. x ph -> A. x A. x ps ) -> A. x ( E. x ph -> A. x ps ) )
5	3 4	syl	|- ( ( E. x ph -> A. x ps ) -> A. x ( E. x ph -> A. x ps ) )