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Theorem enen2 7696
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 7605 . . 3  |-  ( ( C  ~~  A  /\  A  ~~  B )  ->  C  ~~  B )
21ancoms 451 . 2  |-  ( ( A  ~~  B  /\  C  ~~  A )  ->  C  ~~  B )
3 ensym 7602 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
4 entr 7605 . . . 4  |-  ( ( C  ~~  B  /\  B  ~~  A )  ->  C  ~~  A )
54ancoms 451 . . 3  |-  ( ( B  ~~  A  /\  C  ~~  B )  ->  C  ~~  A )
63, 5sylan 469 . 2  |-  ( ( A  ~~  B  /\  C  ~~  B )  ->  C  ~~  A )
72, 6impbida 833 1  |-  ( A 
~~  B  ->  ( C  ~~  A  <->  C  ~~  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   class class class wbr 4395    ~~ cen 7551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-er 7348  df-en 7555
This theorem is referenced by:  karden  8345  ennum  8360  pwcdaen  8597  alephexp1  8986  gchdomtri  9037  gch-kn  9085  ctbnfien  35113
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