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Theorem bj-gl4lem 34603
Description: Lemma for bj-gl4 34604. 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
StepHypRef Expression
1 19.26 1685 . . 3  |-  ( A. x ( A. x ph  /\  ph )  <->  ( A. x A. x ph  /\  A. x ph ) )
2 simpr 459 . . . . 5  |-  ( ( A. x A. x ph  /\  A. x ph )  ->  A. x ph )
32a1i 11 . . . 4  |-  ( ph  ->  ( ( A. x A. x ph  /\  A. x ph )  ->  A. x ph ) )
43anc2ri 556 . . 3  |-  ( ph  ->  ( ( A. x A. x ph  /\  A. x ph )  ->  ( A. x ph  /\  ph ) ) )
51, 4syl5bi 217 . 2  |-  ( ph  ->  ( A. x ( A. x ph  /\  ph )  ->  ( A. x ph  /\  ph )
) )
65alimi 1638 1  |-  ( A. x ph  ->  A. x
( A. x ( A. x ph  /\  ph )  ->  ( A. x ph  /\  ph )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636
This theorem depends on definitions:  df-bi 185  df-an 369
This theorem is referenced by:  bj-gl4  34604
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