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Theorem bnj1023 32129
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1023.1  |-  E. x
( ph  ->  ps )
bnj1023.2  |-  ( ps 
->  ch )
Assertion
Ref Expression
bnj1023  |-  E. x
( ph  ->  ch )

Proof of Theorem bnj1023
StepHypRef Expression
1 bnj1023.2 . . . . 5  |-  ( ps 
->  ch )
21a1i 11 . . . 4  |-  ( (
ph  ->  ps )  -> 
( ps  ->  ch ) )
32ax-gen 1592 . . 3  |-  A. x
( ( ph  ->  ps )  ->  ( ps  ->  ch ) )
4 bnj1023.1 . . 3  |-  E. x
( ph  ->  ps )
5 exintr 1669 . . 3  |-  ( A. x ( ( ph  ->  ps )  ->  ( ps  ->  ch ) )  ->  ( E. x
( ph  ->  ps )  ->  E. x ( (
ph  ->  ps )  /\  ( ps  ->  ch )
) ) )
63, 4, 5mp2 9 . 2  |-  E. x
( ( ph  ->  ps )  /\  ( ps 
->  ch ) )
7 pm3.33 585 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ch ) )  ->  ( ph  ->  ch ) )
86, 7bnj101 32067 1  |-  E. x
( ph  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1368   E.wex 1587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588
This theorem is referenced by:  bnj1098  32132  bnj1110  32328  bnj1118  32330  bnj1128  32336  bnj1145  32339
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