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Theorem bj-axtd 30961
Description: This implication, proved from propositional calculus only (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme  |-  ( A. x ph  ->  ph ) (modal T) implies the axiom scheme  |-  ( A. x ph  ->  E. x ph ) (modal D). See also bj-axdd2 30959 and bj-axd2d 30960. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-axtd  |-  ( ( A. x  -.  ph  ->  -.  ph )  -> 
( ( A. x ph  ->  ph )  ->  ( A. x ph  ->  E. x ph ) ) )

Proof of Theorem bj-axtd
StepHypRef Expression
1 con2 119 . . 3  |-  ( ( A. x  -.  ph  ->  -.  ph )  -> 
( ph  ->  -.  A. x  -.  ph ) )
2 df-ex 1660 . . 3  |-  ( E. x ph  <->  -.  A. x  -.  ph )
31, 2syl6ibr 230 . 2  |-  ( ( A. x  -.  ph  ->  -.  ph )  -> 
( ph  ->  E. x ph ) )
43imim2d 54 1  |-  ( ( A. x  -.  ph  ->  -.  ph )  -> 
( ( A. x ph  ->  ph )  ->  ( A. x ph  ->  E. x ph ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435   E.wex 1659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 188  df-ex 1660
This theorem is referenced by: (None)
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