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Theorem exlimd2 1574
Description: Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1575 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.)
Hypotheses
Ref Expression
exlimd2.1  |-  ( ph  ->  A. x ph )
exlimd2.2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
exlimd2.3  |-  ( ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
exlimd2  |-  ( ph  ->  ( E. x ps 
->  ch ) )

Proof of Theorem exlimd2
StepHypRef Expression
1 exlimd2.1 . . 3  |-  ( ph  ->  A. x ph )
2 exlimd2.2 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
31, 2alrimih 1445 . 2  |-  ( ph  ->  A. x ( ch 
->  A. x ch )
)
4 exlimd2.3 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
51, 4alrimih 1445 . 2  |-  ( ph  ->  A. x ( ps 
->  ch ) )
6 19.23ht 1473 . . 3  |-  ( A. x ( ch  ->  A. x ch )  -> 
( A. x ( ps  ->  ch )  <->  ( E. x ps  ->  ch ) ) )
76biimpd 143 . 2  |-  ( A. x ( ch  ->  A. x ch )  -> 
( A. x ( ps  ->  ch )  ->  ( E. x ps 
->  ch ) ) )
83, 5, 7sylc 62 1  |-  ( ph  ->  ( E. x ps 
->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1329   E.wex 1468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1423  ax-gen 1425  ax-ie2 1470
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  equsexd  1707  cbvexdh  1896
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