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Theorem ndmovdistr 6236
 Description: Any operation is distributive outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
Hypotheses
Ref Expression
ndmov.1
ndmov.5
ndmov.6
Assertion
Ref Expression
ndmovdistr

Proof of Theorem ndmovdistr
StepHypRef Expression
1 ndmov.1 . . . . . . 7
2 ndmov.5 . . . . . . 7
31, 2ndmovrcl 6233 . . . . . 6
43anim2i 553 . . . . 5
5 3anass 940 . . . . 5
64, 5sylibr 204 . . . 4
76con3i 129 . . 3
8 ndmov.6 . . . 4
98ndmov 6231 . . 3
107, 9syl 16 . 2
118, 2ndmovrcl 6233 . . . . . 6
128, 2ndmovrcl 6233 . . . . . 6
1311, 12anim12i 550 . . . . 5
14 anandi 802 . . . . . 6
155, 14bitri 241 . . . . 5
1613, 15sylibr 204 . . . 4
1716con3i 129 . . 3
181ndmov 6231 . . 3
1917, 18syl 16 . 2
2010, 19eqtr4d 2471 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  c0 3628   cxp 4876   cdm 4878  (class class class)co 6081 This theorem is referenced by:  distrpi  8775  distrnq  8838  distrpr  8905  distrsr  8966 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-dm 4888  df-iota 5418  df-fv 5462  df-ov 6084
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