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Theorem bj-axdd2 30731
Description: This implication, proved using only ax-gen 1639 and ax-4 1652 on top of propositional calculus (hence holding, up to the standard interpretation, in any normal modal logic), shows that the axiom scheme  |-  E. x T. implies the axiom scheme 
|-  ( A. x ph  ->  E. x ph ). These correspond to the modal axiom (D), and in predicate calculus, they assert that the universe of discourse is nonempty. For the converse, see bj-axd2d 30732. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-axdd2  |-  ( E. x ph  ->  ( A. x ps  ->  E. x ps ) )

Proof of Theorem bj-axdd2
StepHypRef Expression
1 ala1 1681 . . 3  |-  ( A. x ps  ->  A. x
( ph  ->  ps )
)
2 exim 1675 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
31, 2syl 17 . 2  |-  ( A. x ps  ->  ( E. x ph  ->  E. x ps ) )
43com12 29 1  |-  ( E. x ph  ->  ( A. x ps  ->  E. x ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1403   E.wex 1633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652
This theorem depends on definitions:  df-bi 185  df-ex 1634
This theorem is referenced by: (None)
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