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Theorem entr4i 7562
Description: A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
Hypotheses
Ref Expression
entr4i.1  |-  A  ~~  B
entr4i.2  |-  C  ~~  B
Assertion
Ref Expression
entr4i  |-  A  ~~  C

Proof of Theorem entr4i
StepHypRef Expression
1 entr4i.1 . 2  |-  A  ~~  B
2 entr4i.2 . . 3  |-  C  ~~  B
32ensymi 7555 . 2  |-  B  ~~  C
41, 3entri 7559 1  |-  A  ~~  C
Colors of variables: wff setvar class
Syntax hints:   class class class wbr 4440    ~~ cen 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-er 7301  df-en 7507
This theorem is referenced by:  fodomfi  7788  xpnnen  13792  rpnnen  13810  rexpen  13811  cnso  13830
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