Theorem bj-gl4

Description: In a normal modal logic, the modal axiom GL implies the modal axiom (4). Translated to first-order logic, Axiom GL reads |- ( A. x ( A. x ph -> ph ) -> A. x ph ) . Note that the antecedent of bj-gl4 is an instance of the axiom GL, with ph replaced by ( A. x ph /\ ph ) , which is a modality sometimes called the "strong necessity" of ph . (Contributed by BJ, 12-Dec-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-gl4	|- ( ( A. x ( A. x ( A. x ph /\ ph ) -> ( A. x ph /\ ph ) ) -> A. x ( A. x ph /\ ph ) ) -> ( A. x ph -> A. x A. x ph ) )

Proof

Step	Hyp	Ref	Expression
1		19.26	|- ( A. x ( A. x ph /\ ph ) <-> ( A. x A. x ph /\ A. x ph ) )
2		simpr	|- ( ( A. x A. x ph /\ A. x ph ) -> A. x ph )
3	2	a1i	|- ( ph -> ( ( A. x A. x ph /\ A. x ph ) -> A. x ph ) )
4	3	anc2ri	|- ( ph -> ( ( A. x A. x ph /\ A. x ph ) -> ( A. x ph /\ ph ) ) )
5	1 4	biimtrid	|- ( ph -> ( A. x ( A. x ph /\ ph ) -> ( A. x ph /\ ph ) ) )
6	5	alimi	|- ( A. x ph -> A. x ( A. x ( A. x ph /\ ph ) -> ( A. x ph /\ ph ) ) )
7	1	biimpi	|- ( A. x ( A. x ph /\ ph ) -> ( A. x A. x ph /\ A. x ph ) )
8	6 7	imim12i	|- ( ( A. x ( A. x ( A. x ph /\ ph ) -> ( A. x ph /\ ph ) ) -> A. x ( A. x ph /\ ph ) ) -> ( A. x ph -> ( A. x A. x ph /\ A. x ph ) ) )
9		simpl	|- ( ( A. x A. x ph /\ A. x ph ) -> A. x A. x ph )
10	8 9	syl6	|- ( ( A. x ( A. x ( A. x ph /\ ph ) -> ( A. x ph /\ ph ) ) -> A. x ( A. x ph /\ ph ) ) -> ( A. x ph -> A. x A. x ph ) )