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Theorem cnvin 5984
Description: Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
cnvin (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem cnvin
StepHypRef Expression
1 cnvdif 5983 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴(𝐴𝐵))
2 cnvdif 5983 . . . 4 (𝐴𝐵) = (𝐴𝐵)
32difeq2i 4080 . . 3 (𝐴(𝐴𝐵)) = (𝐴 ∖ (𝐴𝐵))
41, 3eqtri 2847 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴𝐵))
5 dfin4 4227 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
65cnveqi 5726 . 2 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
7 dfin4 4227 . 2 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
84, 6, 73eqtr4i 2857 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cdif 3915  cin 3917  ccnv 5535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110  df-xp 5542  df-rel 5543  df-cnv 5544
This theorem is referenced by:  rnin  5986  dminxp  6018  imainrect  6019  cnvcnv  6030  cnvrescnv  6033  pjdm  20837  ordtrest2  21798  ustexsym  22810  trust  22824  ordtcnvNEW  31181  ordtrest2NEW  31184  msrf  32807  elrn3  33016  pprodcnveq  33362
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