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Theorem r19.2zb 3557
Description: A response to the notion that the condition  A  =/=  (/) can be removed in r19.2z 3556. 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 3556 . . 3  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
21ex 423 . 2  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ph  ->  E. x  e.  A  ph ) )
3 noel 3472 . . . . . . 7  |-  -.  x  e.  (/)
43pm2.21i 123 . . . . . 6  |-  ( x  e.  (/)  ->  ph )
54rgen 2621 . . . . 5  |-  A. x  e.  (/)  ph
6 raleq 2749 . . . . 5  |-  ( A  =  (/)  ->  ( A. x  e.  A  ph  <->  A. x  e.  (/)  ph ) )
75, 6mpbiri 224 . . . 4  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
87necon3bi 2500 . . 3  |-  ( -. 
A. x  e.  A  ph 
->  A  =/=  (/) )
9 exsimpl 1582 . . . 4  |-  ( E. x ( x  e.  A  /\  ph )  ->  E. x  x  e.  A )
10 df-rex 2562 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
11 n0 3477 . . . 4  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
129, 10, 113imtr4i 257 . . 3  |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
138, 12ja 153 . 2  |-  ( ( A. x  e.  A  ph 
->  E. x  e.  A  ph )  ->  A  =/=  (/) )
142, 13impbii 180 1  |-  ( A  =/=  (/)  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   (/)c0 3468
This theorem is referenced by:  iinpreima  5671  intopcoaconlem3  25642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-nul 3469
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