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Theorem r19.2zb 3710
Description: A response to the notion that the condition  A  =/=  (/) can be removed in r19.2z 3709. Interestingly enough,  ph does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)
Assertion
Ref Expression
r19.2zb  |-  ( A  =/=  (/)  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ph ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem r19.2zb
StepHypRef Expression
1 r19.2z 3709 . . 3  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
21ex 424 . 2  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ph  ->  E. x  e.  A  ph ) )
3 noel 3624 . . . . . . 7  |-  -.  x  e.  (/)
43pm2.21i 125 . . . . . 6  |-  ( x  e.  (/)  ->  ph )
54rgen 2763 . . . . 5  |-  A. x  e.  (/)  ph
6 raleq 2896 . . . . 5  |-  ( A  =  (/)  ->  ( A. x  e.  A  ph  <->  A. x  e.  (/)  ph ) )
75, 6mpbiri 225 . . . 4  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
87necon3bi 2639 . . 3  |-  ( -. 
A. x  e.  A  ph 
->  A  =/=  (/) )
9 exsimpl 1602 . . . 4  |-  ( E. x ( x  e.  A  /\  ph )  ->  E. x  x  e.  A )
10 df-rex 2703 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
11 n0 3629 . . . 4  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
129, 10, 113imtr4i 258 . . 3  |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
138, 12ja 155 . 2  |-  ( ( A. x  e.  A  ph 
->  E. x  e.  A  ph )  ->  A  =/=  (/) )
142, 13impbii 181 1  |-  ( A  =/=  (/)  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   (/)c0 3620
This theorem is referenced by:  iinpreima  5852  utopbas  18255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-nul 3621
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