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Theorem enen1 6702
Description: Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
Assertion
Ref Expression
enen1  |-  ( A 
~~  B  ->  ( A  ~~  C  <->  B  ~~  C ) )

Proof of Theorem enen1
StepHypRef Expression
1 ensym 6643 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
2 entr 6646 . . 3  |-  ( ( B  ~~  A  /\  A  ~~  C )  ->  B  ~~  C )
31, 2sylan 281 . 2  |-  ( ( A  ~~  B  /\  A  ~~  C )  ->  B  ~~  C )
4 entr 6646 . 2  |-  ( ( A  ~~  B  /\  B  ~~  C )  ->  A  ~~  C )
53, 4impbida 570 1  |-  ( A 
~~  B  ->  ( A  ~~  C  <->  B  ~~  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   class class class wbr 3899    ~~ cen 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-er 6397  df-en 6603
This theorem is referenced by:  enfi  6735  php5fin  6744  hashen  10498
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