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Theorem entr3i 7478
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 7472 . 2  |-  B  ~~  A
3 entr3i.2 . 2  |-  A  ~~  C
42, 3entri 7476 1  |-  B  ~~  C
Colors of variables: wff setvar class
Syntax hints:   class class class wbr 4403    ~~ cen 7420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-er 7214  df-en 7424
This theorem is referenced by:  xpomenOLD  13614  cpnnen  13632
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