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Theorem bnj1109 28863
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1109.1  |-  E. x
( ( A  =/= 
B  /\  ph )  ->  ps )
bnj1109.2  |-  ( ( A  =  B  /\  ph )  ->  ps )
Assertion
Ref Expression
bnj1109  |-  E. x
( ph  ->  ps )

Proof of Theorem bnj1109
StepHypRef Expression
1 bnj1109.2 . . . . . . 7  |-  ( ( A  =  B  /\  ph )  ->  ps )
21ex 424 . . . . . 6  |-  ( A  =  B  ->  ( ph  ->  ps ) )
32a1i 11 . . . . 5  |-  ( ( A  =/=  B  -> 
( ph  ->  ps )
)  ->  ( A  =  B  ->  ( ph  ->  ps ) ) )
43ax-gen 1552 . . . 4  |-  A. x
( ( A  =/= 
B  ->  ( ph  ->  ps ) )  -> 
( A  =  B  ->  ( ph  ->  ps ) ) )
5 bnj1109.1 . . . . 5  |-  E. x
( ( A  =/= 
B  /\  ph )  ->  ps )
6 impexp 434 . . . . . 6  |-  ( ( ( A  =/=  B  /\  ph )  ->  ps ) 
<->  ( A  =/=  B  ->  ( ph  ->  ps ) ) )
76exbii 1589 . . . . 5  |-  ( E. x ( ( A  =/=  B  /\  ph )  ->  ps )  <->  E. x
( A  =/=  B  ->  ( ph  ->  ps ) ) )
85, 7mpbi 200 . . . 4  |-  E. x
( A  =/=  B  ->  ( ph  ->  ps ) )
9 exintr 1621 . . . 4  |-  ( A. x ( ( A  =/=  B  ->  ( ph  ->  ps ) )  ->  ( A  =  B  ->  ( ph  ->  ps ) ) )  ->  ( E. x
( A  =/=  B  ->  ( ph  ->  ps ) )  ->  E. x
( ( A  =/= 
B  ->  ( ph  ->  ps ) )  /\  ( A  =  B  ->  ( ph  ->  ps ) ) ) ) )
104, 8, 9mp2 9 . . 3  |-  E. x
( ( A  =/= 
B  ->  ( ph  ->  ps ) )  /\  ( A  =  B  ->  ( ph  ->  ps ) ) )
11 exancom 1593 . . 3  |-  ( E. x ( ( A  =/=  B  ->  ( ph  ->  ps ) )  /\  ( A  =  B  ->  ( ph  ->  ps ) ) )  <->  E. x ( ( A  =  B  ->  ( ph  ->  ps ) )  /\  ( A  =/= 
B  ->  ( ph  ->  ps ) ) ) )
1210, 11mpbi 200 . 2  |-  E. x
( ( A  =  B  ->  ( ph  ->  ps ) )  /\  ( A  =/=  B  ->  ( ph  ->  ps ) ) )
13 df-ne 2569 . . . 4  |-  ( A  =/=  B  <->  -.  A  =  B )
1413imbi1i 316 . . 3  |-  ( ( A  =/=  B  -> 
( ph  ->  ps )
)  <->  ( -.  A  =  B  ->  ( ph  ->  ps ) ) )
15 pm2.61 165 . . . 4  |-  ( ( A  =  B  -> 
( ph  ->  ps )
)  ->  ( ( -.  A  =  B  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) ) )
1615imp 419 . . 3  |-  ( ( ( A  =  B  ->  ( ph  ->  ps ) )  /\  ( -.  A  =  B  ->  ( ph  ->  ps ) ) )  -> 
( ph  ->  ps )
)
1714, 16sylan2b 462 . 2  |-  ( ( ( A  =  B  ->  ( ph  ->  ps ) )  /\  ( A  =/=  B  ->  ( ph  ->  ps ) ) )  ->  ( ph  ->  ps ) )
1812, 17bnj101 28794 1  |-  E. x
( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    =/= wne 2567
This theorem is referenced by:  bnj1030  29062  bnj1128  29065  bnj1145  29068
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  df-ne 2569
  Copyright terms: Public domain W3C validator