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Theorem enen1 6734
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 6675 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
2 entr 6678 . . 3  |-  ( ( B  ~~  A  /\  A  ~~  C )  ->  B  ~~  C )
31, 2sylan 281 . 2  |-  ( ( A  ~~  B  /\  A  ~~  C )  ->  B  ~~  C )
4 entr 6678 . 2  |-  ( ( A  ~~  B  /\  B  ~~  C )  ->  A  ~~  C )
53, 4impbida 585 1  |-  ( A 
~~  B  ->  ( A  ~~  C  <->  B  ~~  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   class class class wbr 3929    ~~ cen 6632
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-er 6429  df-en 6635
This theorem is referenced by:  enfi  6767  php5fin  6776  hashen  10537
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