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Theorem cnvin 6144
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 6143 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴(𝐴𝐵))
2 cnvdif 6143 . . . 4 (𝐴𝐵) = (𝐴𝐵)
32difeq2i 4119 . . 3 (𝐴(𝐴𝐵)) = (𝐴 ∖ (𝐴𝐵))
41, 3eqtri 2760 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴𝐵))
5 dfin4 4267 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
65cnveqi 5874 . 2 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
7 dfin4 4267 . 2 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
84, 6, 73eqtr4i 2770 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  cdif 3945  cin 3947  ccnv 5675
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684
This theorem is referenced by:  rnin  6146  dminxp  6179  imainrect  6180  cnvcnv  6191  cnvrescnv  6194  pjdm  21261  ordtrest2  22707  ustexsym  23719  trust  23733  ordtcnvNEW  32895  ordtrest2NEW  32898  msrf  34528  elrn3  34727  pprodcnveq  34850
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