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Theorem enen2 7215
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 7126 . . 3  |-  ( ( C  ~~  A  /\  A  ~~  B )  ->  C  ~~  B )
21ancoms 440 . 2  |-  ( ( A  ~~  B  /\  C  ~~  A )  ->  C  ~~  B )
3 ensym 7123 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
4 entr 7126 . . . 4  |-  ( ( C  ~~  B  /\  B  ~~  A )  ->  C  ~~  A )
54ancoms 440 . . 3  |-  ( ( B  ~~  A  /\  C  ~~  B )  ->  C  ~~  A )
63, 5sylan 458 . 2  |-  ( ( A  ~~  B  /\  C  ~~  B )  ->  C  ~~  A )
72, 6impbida 806 1  |-  ( A 
~~  B  ->  ( C  ~~  A  <->  C  ~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   class class class wbr 4180    ~~ cen 7073
This theorem is referenced by:  karden  7783  ennum  7798  pwcdaen  8029  alephexp1  8418  gchdomtri  8468  gch-kn  8520  ctbnfien  26777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-er 6872  df-en 7077
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