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Theorem ndmovdistr 6464
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 6461 . . . . . 6
43anim2i 569 . . . . 5
5 3anass 977 . . . . 5
64, 5sylibr 212 . . . 4
76con3i 135 . . 3
8 ndmov.6 . . . 4
98ndmov 6459 . . 3
107, 9syl 16 . 2
118, 2ndmovrcl 6461 . . . . . 6
128, 2ndmovrcl 6461 . . . . . 6
1311, 12anim12i 566 . . . . 5
14 anandi3 987 . . . . 5
1513, 14sylibr 212 . . . 4
1615con3i 135 . . 3
171ndmov 6459 . . 3
1816, 17syl 16 . 2
1910, 18eqtr4d 2501 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818   c0 3784  X.cxp 5002  domcdm 5004  (class class class)co 6296
This theorem is referenced by:  distrpi  9297  distrnq  9360  distrpr  9427  distrsr  9489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-xp 5010  df-dm 5014  df-iota 5556  df-fv 5601  df-ov 6299
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