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Theorem entr3i 7583
Description: A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
Hypotheses
Ref Expression
entr3i.1  |-  A  ~~  B
entr3i.2  |-  A  ~~  C
Assertion
Ref Expression
entr3i  |-  B  ~~  C

Proof of Theorem entr3i
StepHypRef Expression
1 entr3i.1 . . 3  |-  A  ~~  B
21ensymi 7577 . 2  |-  B  ~~  A
3 entr3i.2 . 2  |-  A  ~~  C
42, 3entri 7581 1  |-  B  ~~  C
Colors of variables: wff setvar class
Syntax hints:   class class class wbr 4453    ~~ cen 7525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-er 7323  df-en 7529
This theorem is referenced by:  xpomenOLD  13822  cpnnen  13840
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