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Theorem r19.29imd 2991
Description: Theorem 19.29 of [Margaris] p. 90 with an implication in the hypothesis containing the generalization, deduction version. (Contributed by AV, 19-Jan-2019.)
Hypotheses
Ref Expression
r19.29imd.1  |-  ( ph  ->  E. x  e.  A  ps )
r19.29imd.2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
Assertion
Ref Expression
r19.29imd  |-  ( ph  ->  E. x  e.  A  ( ps  /\  ch )
)

Proof of Theorem r19.29imd
StepHypRef Expression
1 r19.29imd.1 . . 3  |-  ( ph  ->  E. x  e.  A  ps )
2 r19.29imd.2 . . 3  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
3 r19.29r 2990 . . 3  |-  ( ( E. x  e.  A  ps  /\  A. x  e.  A  ( ps  ->  ch ) )  ->  E. x  e.  A  ( ps  /\  ( ps  ->  ch ) ) )
41, 2, 3syl2anc 659 . 2  |-  ( ph  ->  E. x  e.  A  ( ps  /\  ( ps  ->  ch ) ) )
5 abai 793 . . 3  |-  ( ( ps  /\  ch )  <->  ( ps  /\  ( ps 
->  ch ) ) )
65rexbii 2956 . 2  |-  ( E. x  e.  A  ( ps  /\  ch )  <->  E. x  e.  A  ( ps  /\  ( ps 
->  ch ) ) )
74, 6sylibr 212 1  |-  ( ph  ->  E. x  e.  A  ( ps  /\  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wral 2804   E.wrex 2805
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  df-ex 1618  df-ral 2809  df-rex 2810
This theorem is referenced by:  psgndif  18811
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