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Theorem r19.2zb 3918
Description: A response to the notion that the condition  A  =/=  (/) can be removed in r19.2z 3917. 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 3917 . . 3  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
21ex 434 . 2  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ph  ->  E. x  e.  A  ph ) )
3 noel 3789 . . . . . . 7  |-  -.  x  e.  (/)
43pm2.21i 131 . . . . . 6  |-  ( x  e.  (/)  ->  ph )
54rgen 2824 . . . . 5  |-  A. x  e.  (/)  ph
6 raleq 3058 . . . . 5  |-  ( A  =  (/)  ->  ( A. x  e.  A  ph  <->  A. x  e.  (/)  ph ) )
75, 6mpbiri 233 . . . 4  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
87necon3bi 2696 . . 3  |-  ( -. 
A. x  e.  A  ph 
->  A  =/=  (/) )
9 exsimpl 1654 . . . 4  |-  ( E. x ( x  e.  A  /\  ph )  ->  E. x  x  e.  A )
10 df-rex 2820 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
11 n0 3794 . . . 4  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
129, 10, 113imtr4i 266 . . 3  |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
138, 12ja 161 . 2  |-  ( ( A. x  e.  A  ph 
->  E. x  e.  A  ph )  ->  A  =/=  (/) )
142, 13impbii 188 1  |-  ( A  =/=  (/)  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-nul 3786
This theorem is referenced by:  iinpreima  6012  utopbas  20565
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