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Theorem bnj1023 28489
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 1552 . . 3  |-  A. x
( ( ph  ->  ps )  ->  ( ps  ->  ch ) )
4 bnj1023.1 . . 3  |-  E. x
( ph  ->  ps )
5 exintr 1621 . . 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 pm3.33 569 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ch ) )  ->  ( ph  ->  ch ) )
86, 7bnj101 28426 1  |-  E. x
( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547
This theorem is referenced by:  bnj1098  28492  bnj1110  28689  bnj1118  28691  bnj1128  28697  bnj1145  28700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548
  Copyright terms: Public domain W3C validator