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Theorem acongid 26892
 Description: A wff like that in this theorem will be known as an "alternating congruence". A special symbol might be considered if more uses come up. They have many of the same properties as normal congruences, starting with reflexivity. JonesMatijasevic uses "a ≡ ± b (mod c)" for this construction. The disjunction of divisibility constraints seems to adequately capture the concept, but it's rather verbose and somewhat inelegant. Use of an explicit equivalence relation might also work. (Contributed by Stefan O'Rear, 2-Oct-2014.)
Assertion
Ref Expression
acongid

Proof of Theorem acongid
StepHypRef Expression
1 congid 26888 . 2
21orcd 382 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 358   wa 359   wcel 1721   class class class wbr 4167  (class class class)co 6034   cmin 9237  cneg 9238  cz 10228   cdivides 12793 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  ax-resscn 8994  ax-1cn 8995  ax-icn 8996  ax-addcl 8997  ax-addrcl 8998  ax-mulcl 8999  ax-mulrcl 9000  ax-mulcom 9001  ax-addass 9002  ax-mulass 9003  ax-distr 9004  ax-i2m1 9005  ax-1ne0 9006  ax-1rid 9007  ax-rnegex 9008  ax-rrecex 9009  ax-cnre 9010  ax-pre-lttri 9011  ax-pre-lttrn 9012  ax-pre-ltadd 9013 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-nel 2567  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-nul 3586  df-if 3697  df-pw 3758  df-sn 3777  df-pr 3778  df-op 3780  df-uni 3972  df-br 4168  df-opab 4222  df-mpt 4223  df-id 4453  df-po 4458  df-so 4459  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-riota 6499  df-er 6855  df-en 7060  df-dom 7061  df-sdom 7062  df-pnf 9069  df-mnf 9070  df-ltxr 9072  df-sub 9239  df-neg 9240  df-z 10229  df-dvds 12794
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