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Theorem eqord1 9547
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 9546 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <_  D  <->  M  <_  N ) )
81, 3, 2, 4, 5, 6leord1 9546 . . . 4  |-  ( (
ph  /\  ( D  e.  S  /\  C  e.  S ) )  -> 
( D  <_  C  <->  N  <_  M ) )
98ancom2s 778 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( D  <_  C  <->  N  <_  M ) )
107, 9anbi12d 692 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ( C  <_  D  /\  D  <_  C
)  <->  ( M  <_  N  /\  N  <_  M
) ) )
114sseli 3336 . . . 4  |-  ( C  e.  S  ->  C  e.  RR )
124sseli 3336 . . . 4  |-  ( D  e.  S  ->  D  e.  RR )
13 letri3 9152 . . . 4  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  =  D  <-> 
( C  <_  D  /\  D  <_  C ) ) )
1411, 12, 13syl2an 464 . . 3  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( C  =  D  <-> 
( C  <_  D  /\  D  <_  C ) ) )
1514adantl 453 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <-> 
( C  <_  D  /\  D  <_  C ) ) )
165ralrimiva 2781 . . . . 5  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
172eleq1d 2501 . . . . . 6  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
1817rspccva 3043 . . . . 5  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
1916, 18sylan 458 . . . 4  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
2019adantrr 698 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  RR )
213eleq1d 2501 . . . . . 6  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2221rspccva 3043 . . . . 5  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2316, 22sylan 458 . . . 4  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2423adantrl 697 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  RR )
2520, 24letri3d 9207 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
2610, 15, 253bitr4d 277 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <-> 
M  =  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   class class class wbr 4204   RRcr 8981    < clt 9112    <_ cle 9113
This theorem is referenced by:  eqord2  9550  expcan  11424  ovolicc2lem3  19407  rmyeq0  27009  rmyeq  27010
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118
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