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Theorem enqeq 9635
 Description: Corollary of nqereu 9630: 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 1051 . 2 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → (𝐴Q𝐵Q))
2 elpqn 9626 . . . . 5 (𝐵Q𝐵 ∈ (N × N))
323ad2ant2 1076 . . . 4 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → 𝐵 ∈ (N × N))
4 nqereu 9630 . . . 4 (𝐵 ∈ (N × N) → ∃!𝑥Q 𝑥 ~Q 𝐵)
5 reurmo 3138 . . . 4 (∃!𝑥Q 𝑥 ~Q 𝐵 → ∃*𝑥Q 𝑥 ~Q 𝐵)
63, 4, 53syl 18 . . 3 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → ∃*𝑥Q 𝑥 ~Q 𝐵)
7 df-rmo 2904 . . 3 (∃*𝑥Q 𝑥 ~Q 𝐵 ↔ ∃*𝑥(𝑥Q𝑥 ~Q 𝐵))
86, 7sylib 207 . 2 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → ∃*𝑥(𝑥Q𝑥 ~Q 𝐵))
9 3simpb 1052 . 2 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → (𝐴Q𝐴 ~Q 𝐵))
10 simp2 1055 . . 3 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → 𝐵Q)
11 enqer 9622 . . . . 5 ~Q Er (N × N)
1211a1i 11 . . . 4 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → ~Q Er (N × N))
1312, 3erref 7649 . . 3 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → 𝐵 ~Q 𝐵)
1410, 13jca 553 . 2 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → (𝐵Q𝐵 ~Q 𝐵))
15 eleq1 2676 . . . 4 (𝑥 = 𝐴 → (𝑥Q𝐴Q))
16 breq1 4586 . . . 4 (𝑥 = 𝐴 → (𝑥 ~Q 𝐵𝐴 ~Q 𝐵))
1715, 16anbi12d 743 . . 3 (𝑥 = 𝐴 → ((𝑥Q𝑥 ~Q 𝐵) ↔ (𝐴Q𝐴 ~Q 𝐵)))
18 eleq1 2676 . . . 4 (𝑥 = 𝐵 → (𝑥Q𝐵Q))
19 breq1 4586 . . . 4 (𝑥 = 𝐵 → (𝑥 ~Q 𝐵𝐵 ~Q 𝐵))
2018, 19anbi12d 743 . . 3 (𝑥 = 𝐵 → ((𝑥Q𝑥 ~Q 𝐵) ↔ (𝐵Q𝐵 ~Q 𝐵)))
2117, 20moi 3356 . 2 (((𝐴Q𝐵Q) ∧ ∃*𝑥(𝑥Q𝑥 ~Q 𝐵) ∧ ((𝐴Q𝐴 ~Q 𝐵) ∧ (𝐵Q𝐵 ~Q 𝐵))) → 𝐴 = 𝐵)
221, 8, 9, 14, 21syl112anc 1322 1 ((𝐴Q𝐵Q𝐴 ~Q 𝐵) → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459  ∃!wreu 2898  ∃*wrmo 2899   class class class wbr 4583   × cxp 5036   Er wer 7626  Ncnpi 9545   ~Q ceq 9552  Qcnq 9553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-mi 9575  df-lti 9576  df-enq 9612  df-nq 9613 This theorem is referenced by:  nqereq  9636  ltsonq  9670
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