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Theorem enen2 6864
Description: Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
Assertion
Ref Expression
enen2  |-  ( A 
~~  B  ->  ( C  ~~  A  <->  C  ~~  B ) )

Proof of Theorem enen2
StepHypRef Expression
1 entr 6805 . . 3  |-  ( ( C  ~~  A  /\  A  ~~  B )  ->  C  ~~  B )
21ancoms 268 . 2  |-  ( ( A  ~~  B  /\  C  ~~  A )  ->  C  ~~  B )
3 ensym 6802 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
4 entr 6805 . . . 4  |-  ( ( C  ~~  B  /\  B  ~~  A )  ->  C  ~~  A )
54ancoms 268 . . 3  |-  ( ( B  ~~  A  /\  C  ~~  B )  ->  C  ~~  A )
63, 5sylan 283 . 2  |-  ( ( A  ~~  B  /\  C  ~~  B )  ->  C  ~~  A )
72, 6impbida 596 1  |-  ( A 
~~  B  ->  ( C  ~~  A  <->  C  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   class class class wbr 4018    ~~ cen 6759
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-er 6554  df-en 6762
This theorem is referenced by:  php5fin  6905  carden2bex  7213  hashen  10791
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