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Theorem cnvdif 5270
 Description: Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
cnvdif

Proof of Theorem cnvdif
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5234 . 2
2 difss 3466 . . 3
3 relcnv 5234 . . 3
4 relss 4955 . . 3
52, 3, 4mp2 9 . 2
6 eldif 3322 . . 3
7 vex 2951 . . . 4
8 vex 2951 . . . 4
97, 8opelcnv 5046 . . 3
10 eldif 3322 . . . 4
117, 8opelcnv 5046 . . . . 5
127, 8opelcnv 5046 . . . . . 6
1312notbii 288 . . . . 5
1411, 13anbi12i 679 . . . 4
1510, 14bitri 241 . . 3
166, 9, 153bitr4i 269 . 2
171, 5, 16eqrelriiv 4962 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 359   wceq 1652   wcel 1725   cdif 3309   wss 3312  cop 3809  ccnv 4869   wrel 4875 This theorem is referenced by:  cnvin  5271  gtiso  24080 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878
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