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Theorem ndmovord 6350
Description: Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.)
Hypotheses
Ref Expression
ndmov.1  |-  dom  F  =  ( S  X.  S )
ndmovord.4  |-  R  C_  ( S  X.  S
)
ndmovord.5  |-  -.  (/)  e.  S
ndmovord.6  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( A R B  <-> 
( C F A ) R ( C F B ) ) )
Assertion
Ref Expression
ndmovord  |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )

Proof of Theorem ndmovord
StepHypRef Expression
1 ndmovord.6 . . 3  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( A R B  <-> 
( C F A ) R ( C F B ) ) )
213expia 1190 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( C  e.  S  ->  ( A R B  <-> 
( C F A ) R ( C F B ) ) ) )
3 ndmovord.4 . . . . 5  |-  R  C_  ( S  X.  S
)
43brel 4982 . . . 4  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
53brel 4982 . . . . 5  |-  ( ( C F A ) R ( C F B )  ->  (
( C F A )  e.  S  /\  ( C F B )  e.  S ) )
6 ndmov.1 . . . . . . . 8  |-  dom  F  =  ( S  X.  S )
7 ndmovord.5 . . . . . . . 8  |-  -.  (/)  e.  S
86, 7ndmovrcl 6346 . . . . . . 7  |-  ( ( C F A )  e.  S  ->  ( C  e.  S  /\  A  e.  S )
)
98simprd 463 . . . . . 6  |-  ( ( C F A )  e.  S  ->  A  e.  S )
106, 7ndmovrcl 6346 . . . . . . 7  |-  ( ( C F B )  e.  S  ->  ( C  e.  S  /\  B  e.  S )
)
1110simprd 463 . . . . . 6  |-  ( ( C F B )  e.  S  ->  B  e.  S )
129, 11anim12i 566 . . . . 5  |-  ( ( ( C F A )  e.  S  /\  ( C F B )  e.  S )  -> 
( A  e.  S  /\  B  e.  S
) )
135, 12syl 16 . . . 4  |-  ( ( C F A ) R ( C F B )  ->  ( A  e.  S  /\  B  e.  S )
)
144, 13pm5.21ni 352 . . 3  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
1514a1d 25 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) ) )
162, 15pm2.61i 164 1  |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3423   (/)c0 3732   class class class wbr 4387    X. cxp 4933   dom cdm 4935  (class class class)co 6187
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 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-xp 4941  df-dm 4945  df-iota 5476  df-fv 5521  df-ov 6190
This theorem is referenced by:  ltapi  9170  ltmpi  9171  ltanq  9238  ltmnq  9239  ltapr  9312  ltasr  9365
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