Theorem bj-nnftht

Description: A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp (modal T), as in bj-nnfbi . (Contributed by BJ, 28-Jul-2023)

	Ref	Expression
Assertion	bj-nnftht	|- ( ( ph /\ A. x ph ) -> F// x ph )

Proof

Step	Hyp	Ref	Expression
1		ax-1	|- ( ph -> ( E. x ph -> ph ) )
2		ax-1	|- ( A. x ph -> ( ph -> A. x ph ) )
3	1 2	anim12i	|- ( ( ph /\ A. x ph ) -> ( ( E. x ph -> ph ) /\ ( ph -> A. x ph ) ) )
4		df-bj-nnf	|- ( F// x ph <-> ( ( E. x ph -> ph ) /\ ( ph -> A. x ph ) ) )
5	3 4	sylibr	|- ( ( ph /\ A. x ph ) -> F// x ph )