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Theorem cnegex2 8110
Description: Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
cnegex2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
Distinct variable group:    x, A

Proof of Theorem cnegex2
StepHypRef Expression
1 cnegex 8109 . 2  |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
2 addcom 8068 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  +  x
)  =  ( x  +  A ) )
32eqeq1d 2184 . . 3  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( ( A  +  x )  =  0  <-> 
( x  +  A
)  =  0 ) )
43rexbidva 2472 . 2  |-  ( A  e.  CC  ->  ( E. x  e.  CC  ( A  +  x
)  =  0  <->  E. x  e.  CC  (
x  +  A )  =  0 ) )
51, 4mpbid 148 1  |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2146   E.wrex 2454  (class class class)co 5865   CCcc 7784   0cc0 7786    + caddc 7789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-resscn 7878  ax-1cn 7879  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868
This theorem is referenced by:  addcan  8111
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