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Theorem r19.2zb 3832
Description: A response to the notion that the condition  A  =/=  (/) can be removed in r19.2z 3831. 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 3831 . . 3  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
21ex 435 . 2  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ph  ->  E. x  e.  A  ph ) )
3 noel 3708 . . . . . . 7  |-  -.  x  e.  (/)
43pm2.21i 134 . . . . . 6  |-  ( x  e.  (/)  ->  ph )
54rgen 2724 . . . . 5  |-  A. x  e.  (/)  ph
6 raleq 2964 . . . . 5  |-  ( A  =  (/)  ->  ( A. x  e.  A  ph  <->  A. x  e.  (/)  ph ) )
75, 6mpbiri 236 . . . 4  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
87necon3bi 2627 . . 3  |-  ( -. 
A. x  e.  A  ph 
->  A  =/=  (/) )
9 exsimpl 1723 . . . 4  |-  ( E. x ( x  e.  A  /\  ph )  ->  E. x  x  e.  A )
10 df-rex 2720 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
11 n0 3714 . . . 4  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
129, 10, 113imtr4i 269 . . 3  |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
138, 12ja 164 . 2  |-  ( ( A. x  e.  A  ph 
->  E. x  e.  A  ph )  ->  A  =/=  (/) )
142, 13impbii 190 1  |-  ( A  =/=  (/)  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2599   A.wral 2714   E.wrex 2715   (/)c0 3704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-v 3024  df-dif 3382  df-nul 3705
This theorem is referenced by:  iinpreima  5969  utopbas  21192  radcnvrat  36576
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