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Theorem enen1 7249
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 7158 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
2 entr 7161 . . 3  |-  ( ( B  ~~  A  /\  A  ~~  C )  ->  B  ~~  C )
31, 2sylan 459 . 2  |-  ( ( A  ~~  B  /\  A  ~~  C )  ->  B  ~~  C )
4 entr 7161 . 2  |-  ( ( A  ~~  B  /\  B  ~~  C )  ->  A  ~~  C )
53, 4impbida 807 1  |-  ( A 
~~  B  ->  ( A  ~~  C  <->  B  ~~  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   class class class wbr 4214    ~~ cen 7108
This theorem is referenced by:  onomeneq  7298  enfi  7327  alephexp2  8458  pmtrfmvdn0  27382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-er 6907  df-en 7112
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