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Theorem enqeq 10348
Description: Corollary of nqereu 10343: 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 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → 𝐴 = 𝐵)

Proof of Theorem enqeq
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 3simpa 1142 . 2 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → (𝐴Q𝐵Q))
2 elpqn 10339 . . . . 5 (𝐵Q𝐵 ∈ (N × N))
323ad2ant2 1128 . . . 4 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → 𝐵 ∈ (N × N))
4 nqereu 10343 . . . 4 (𝐵 ∈ (N × N) → ∃!𝑥Q 𝑥 ~Q 𝐵)
5 reurmo 3438 . . . 4 (∃!𝑥Q 𝑥 ~Q 𝐵 → ∃*𝑥Q 𝑥 ~Q 𝐵)
63, 4, 53syl 18 . . 3 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → ∃*𝑥Q 𝑥 ~Q 𝐵)
7 df-rmo 3150 . . 3 (∃*𝑥Q 𝑥 ~Q 𝐵 ↔ ∃*𝑥(𝑥Q𝑥 ~Q 𝐵))
86, 7sylib 219 . 2 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → ∃*𝑥(𝑥Q𝑥 ~Q 𝐵))
9 3simpb 1143 . 2 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → (𝐴Q𝐴 ~Q 𝐵))
10 simp2 1131 . . 3 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → 𝐵Q)
11 enqer 10335 . . . . 5 ~Q Er (N × N)
1211a1i 11 . . . 4 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → ~Q Er (N × N))
1312, 3erref 8302 . . 3 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → 𝐵 ~Q 𝐵)
1410, 13jca 512 . 2 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → (𝐵Q𝐵 ~Q 𝐵))
15 eleq1 2904 . . . 4 (𝑥 = 𝐴 → (𝑥Q𝐴Q))
16 breq1 5065 . . . 4 (𝑥 = 𝐴 → (𝑥 ~Q 𝐵𝐴 ~Q 𝐵))
1715, 16anbi12d 630 . . 3 (𝑥 = 𝐴 → ((𝑥Q𝑥 ~Q 𝐵) ↔ (𝐴Q𝐴 ~Q 𝐵)))
18 eleq1 2904 . . . 4 (𝑥 = 𝐵 → (𝑥Q𝐵Q))
19 breq1 5065 . . . 4 (𝑥 = 𝐵 → (𝑥 ~Q 𝐵𝐵 ~Q 𝐵))
2018, 19anbi12d 630 . . 3 (𝑥 = 𝐵 → ((𝑥Q𝑥 ~Q 𝐵) ↔ (𝐵Q𝐵 ~Q 𝐵)))
2117, 20moi 3712 . 2 (((𝐴Q𝐵Q) ∧ ∃*𝑥(𝑥Q𝑥 ~Q 𝐵) ∧ ((𝐴Q𝐴 ~Q 𝐵) ∧ (𝐵Q𝐵 ~Q 𝐵))) → 𝐴 = 𝐵)
221, 8, 9, 14, 21syl112anc 1368 1 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  ∃*wmo 2617  ∃!wreu 3144  ∃*wrmo 3145   class class class wbr 5062   × cxp 5551   Er wer 8279  Ncnpi 10258   ~Q ceq 10265  Qcnq 10266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-oadd 8100  df-omul 8101  df-er 8282  df-ni 10286  df-mi 10288  df-lti 10289  df-enq 10325  df-nq 10326
This theorem is referenced by:  nqereq  10349  ltsonq  10383
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