Theorem bj-wnfnf

Description: When ph is substituted for ps , this statement expresses nonfreeness in the weak form of nonfreeness ( E. -> A. ) . Note that this could also be proved from bj-nnfim , bj-nnfe1 and bj-nnfa1 . (Contributed by BJ, 9-Dec-2023)

	Ref	Expression
Assertion	bj-wnfnf	|- F// x ( E. x ph -> A. x ps )

Proof

Step	Hyp	Ref	Expression
1		bj-wnf2	|- ( E. x ( E. x ph -> A. x ps ) -> ( E. x ph -> A. x ps ) )
2		bj-wnf1	|- ( ( E. x ph -> A. x ps ) -> A. x ( E. x ph -> A. x ps ) )
3		df-bj-nnf	|- ( F// x ( E. x ph -> A. x ps ) <-> ( ( E. x ( E. x ph -> A. x ps ) -> ( E. x ph -> A. x ps ) ) /\ ( ( E. x ph -> A. x ps ) -> A. x ( E. x ph -> A. x ps ) ) ) )
4	1 2 3	mpbir2an	|- F// x ( E. x ph -> A. x ps )