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Theorem eqord1 9346
Description: Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ltord.1  |-  ( x  =  y  ->  A  =  B )
ltord.2  |-  ( x  =  C  ->  A  =  M )
ltord.3  |-  ( x  =  D  ->  A  =  N )
ltord.4  |-  S  C_  RR
ltord.5  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  RR )
ltord.6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  ->  A  <  B ) )
Assertion
Ref Expression
eqord1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <-> 
M  =  N ) )
Distinct variable groups:    x, B    x, y, C    x, D, y    x, M, y    x, N, y    ph, x, y   
x, S, y
Allowed substitution hints:    A( x, y)    B( y)

Proof of Theorem eqord1
StepHypRef Expression
1 ltord.1 . . . 4  |-  ( x  =  y  ->  A  =  B )
2 ltord.2 . . . 4  |-  ( x  =  C  ->  A  =  M )
3 ltord.3 . . . 4  |-  ( x  =  D  ->  A  =  N )
4 ltord.4 . . . 4  |-  S  C_  RR
5 ltord.5 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  RR )
6 ltord.6 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  ->  A  <  B ) )
71, 2, 3, 4, 5, 6leord1 9345 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <_  D  <->  M  <_  N ) )
81, 3, 2, 4, 5, 6leord1 9345 . . . 4  |-  ( (
ph  /\  ( D  e.  S  /\  C  e.  S ) )  -> 
( D  <_  C  <->  N  <_  M ) )
98ancom2s 777 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( D  <_  C  <->  N  <_  M ) )
107, 9anbi12d 691 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ( C  <_  D  /\  D  <_  C
)  <->  ( M  <_  N  /\  N  <_  M
) ) )
114sseli 3210 . . . 4  |-  ( C  e.  S  ->  C  e.  RR )
124sseli 3210 . . . 4  |-  ( D  e.  S  ->  D  e.  RR )
13 letri3 8952 . . . 4  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  =  D  <-> 
( C  <_  D  /\  D  <_  C ) ) )
1411, 12, 13syl2an 463 . . 3  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( C  =  D  <-> 
( C  <_  D  /\  D  <_  C ) ) )
1514adantl 452 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <-> 
( C  <_  D  /\  D  <_  C ) ) )
165ralrimiva 2660 . . . . 5  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
172eleq1d 2382 . . . . . 6  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
1817rspccva 2917 . . . . 5  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
1916, 18sylan 457 . . . 4  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
2019adantrr 697 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  RR )
213eleq1d 2382 . . . . . 6  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2221rspccva 2917 . . . . 5  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2316, 22sylan 457 . . . 4  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2423adantrl 696 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  RR )
2520, 24letri3d 9006 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
2610, 15, 253bitr4d 276 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <-> 
M  =  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577    C_ wss 3186   class class class wbr 4060   RRcr 8781    < clt 8912    <_ cle 8913
This theorem is referenced by:  eqord2  9349  expcan  11201  ovolicc2lem3  18931  rmyeq0  26188  rmyeq  26189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-resscn 8839  ax-pre-lttri 8856  ax-pre-lttrn 8857
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918
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