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Theorem ndmovord 6446
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 1197 . 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 5034 . . . 4  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
53brel 5034 . . . . 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 6442 . . . . . . 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 6442 . . . . . . 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 972    = wceq 1381    e. wcel 1802    C_ wss 3458   (/)c0 3767   class class class wbr 4433    X. cxp 4983   dom cdm 4985  (class class class)co 6277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-xp 4991  df-dm 4995  df-iota 5537  df-fv 5582  df-ov 6280
This theorem is referenced by:  ltapi  9279  ltmpi  9280  ltanq  9347  ltmnq  9348  ltapr  9421  ltasr  9475
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