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Theorem enqer 8745
 Description: The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.)
Assertion
Ref Expression
enqer

Proof of Theorem enqer
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-enq 8735 . 2
2 mulcompi 8720 . 2
3 mulclpi 8717 . 2
4 mulasspi 8721 . 2
5 mulcanpi 8724 . . 3
65biimpd 199 . 2
71, 2, 3, 4, 6ecopover 6958 1
 Colors of variables: wff set class Syntax hints:   wa 359   wceq 1649   wcel 1721   cxp 4830  (class class class)co 6034   wer 6852  cnpi 8666   cmi 8668   ceq 8673 This theorem is referenced by:  nqereu  8753  nqerf  8754  nqerid  8757  enqeq  8758  nqereq  8759  adderpq  8780  mulerpq  8781  1nqenq  8786 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2382  ax-sep 4285  ax-nul 4293  ax-pow 4332  ax-pr 4358  ax-un 4655 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2256  df-mo 2257  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2526  df-ne 2566  df-ral 2668  df-rex 2669  df-reu 2670  df-rab 2672  df-v 2915  df-sbc 3119  df-csb 3209  df-dif 3280  df-un 3282  df-in 3284  df-ss 3291  df-pss 3293  df-nul 3586  df-if 3697  df-pw 3758  df-sn 3777  df-pr 3778  df-tp 3779  df-op 3780  df-uni 3972  df-iun 4051  df-br 4168  df-opab 4222  df-mpt 4223  df-tr 4258  df-eprel 4449  df-id 4453  df-po 4458  df-so 4459  df-fr 4496  df-we 4498  df-ord 4539  df-on 4540  df-lim 4541  df-suc 4542  df-om 4800  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5372  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-ov 6037  df-oprab 6038  df-mpt2 6039  df-1st 6302  df-2nd 6303  df-recs 6583  df-rdg 6618  df-oadd 6678  df-omul 6679  df-er 6855  df-ni 8696  df-mi 8698  df-enq 8735
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