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Theorem enqeq 8558
Description: Corollary of nqereu 8553: if two fractions are both reduced and equivalent, then they are equal. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
enqeq  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  A  ~Q  B )  ->  A  =  B )

Proof of Theorem enqeq
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 3simpa 952 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  A  ~Q  B )  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
2 elpqn 8549 . . . . 5  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
323ad2ant2 977 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  A  ~Q  B )  ->  B  e.  ( N.  X.  N. ) )
4 nqereu 8553 . . . 4  |-  ( B  e.  ( N.  X.  N. )  ->  E! x  e.  Q.  x  ~Q  B
)
5 reu5 2753 . . . . 5  |-  ( E! x  e.  Q.  x  ~Q  B  <->  ( E. x  e.  Q.  x  ~Q  B  /\  E* x  e.  Q. x  ~Q  B ) )
65simprbi 450 . . . 4  |-  ( E! x  e.  Q.  x  ~Q  B  ->  E* x  e.  Q. x  ~Q  B
)
73, 4, 63syl 18 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  A  ~Q  B )  ->  E* x  e.  Q. x  ~Q  B )
8 df-rmo 2551 . . 3  |-  ( E* x  e.  Q. x  ~Q  B  <->  E* x ( x  e.  Q.  /\  x  ~Q  B ) )
97, 8sylib 188 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  A  ~Q  B )  ->  E* x ( x  e. 
Q.  /\  x  ~Q  B ) )
10 3simpb 953 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  A  ~Q  B )  ->  ( A  e.  Q.  /\  A  ~Q  B ) )
11 simp2 956 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  A  ~Q  B )  ->  B  e.  Q. )
12 enqer 8545 . . . . 5  |-  ~Q  Er  ( N.  X.  N. )
1312a1i 10 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  A  ~Q  B )  ->  ~Q  Er  ( N.  X.  N. )
)
1413, 3erref 6680 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  A  ~Q  B )  ->  B  ~Q  B )
1511, 14jca 518 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  A  ~Q  B )  ->  ( B  e.  Q.  /\  B  ~Q  B ) )
16 eleq1 2343 . . . 4  |-  ( x  =  A  ->  (
x  e.  Q.  <->  A  e.  Q. ) )
17 breq1 4026 . . . 4  |-  ( x  =  A  ->  (
x  ~Q  B  <->  A  ~Q  B ) )
1816, 17anbi12d 691 . . 3  |-  ( x  =  A  ->  (
( x  e.  Q.  /\  x  ~Q  B )  <-> 
( A  e.  Q.  /\  A  ~Q  B ) ) )
19 eleq1 2343 . . . 4  |-  ( x  =  B  ->  (
x  e.  Q.  <->  B  e.  Q. ) )
20 breq1 4026 . . . 4  |-  ( x  =  B  ->  (
x  ~Q  B  <->  B  ~Q  B ) )
2119, 20anbi12d 691 . . 3  |-  ( x  =  B  ->  (
( x  e.  Q.  /\  x  ~Q  B )  <-> 
( B  e.  Q.  /\  B  ~Q  B ) ) )
2218, 21moi 2948 . 2  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  E* x ( x  e.  Q.  /\  x  ~Q  B )  /\  (
( A  e.  Q.  /\  A  ~Q  B )  /\  ( B  e. 
Q.  /\  B  ~Q  B ) ) )  ->  A  =  B )
231, 9, 10, 15, 22syl112anc 1186 1  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  A  ~Q  B )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E*wmo 2144   E.wrex 2544   E!wreu 2545   E*wrmo 2546   class class class wbr 4023    X. cxp 4687    Er wer 6657   N.cnpi 8466    ~Q ceq 8473   Q.cnq 8474
This theorem is referenced by:  nqereq  8559  ltsonq  8593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-mi 8498  df-lti 8499  df-enq 8535  df-nq 8536
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