Theorem bj-gl4lem 32579

Description: Lemma for bj-gl4 32580. Note that this proof holds in the modal logic (K). (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-gl4lem	|- ( A. x ph -> A. x ( A. x ( A. x ph /\ ph ) -> ( A. x ph /\ ph ) ) )

Proof of Theorem bj-gl4lem

Step	Hyp	Ref	Expression
1		19.26 1798	. . 3 |- ( A. x ( A. x ph /\ ph ) <-> ( A. x A. x ph /\ A. x ph ) )
2		simpr 477	. . . . 5 |- ( ( A. x A. x ph /\ A. x ph ) -> A. x ph )
3	2	a1i 11	. . . 4 |- ( ph -> ( ( A. x A. x ph /\ A. x ph ) -> A. x ph ) )
4	3	anc2ri 581	. . 3 |- ( ph -> ( ( A. x A. x ph /\ A. x ph ) -> ( A. x ph /\ ph ) ) )
5	1, 4	syl5bi 232	. 2 |- ( ph -> ( A. x ( A. x ph /\ ph ) -> ( A. x ph /\ ph ) ) )
6	5	alimi 1739	1 |- ( A. x ph -> A. x ( A. x ( A. x ph /\ ph ) -> ( A. x ph /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by:  bj-gl4  32580