Theorem cnvin 6037

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	⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)

Proof of Theorem cnvin

Step	Hyp	Ref	Expression
1		cnvdif 6036	. . 3 ⊢ ◡(𝐴 ∖ (𝐴 ∖ 𝐵)) = (◡𝐴 ∖ ◡(𝐴 ∖ 𝐵))
2		cnvdif 6036	. . . 4 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵)
3	2	difeq2i 4050	. . 3 ⊢ (◡𝐴 ∖ ◡(𝐴 ∖ 𝐵)) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵))
4	1, 3	eqtri 2766	. 2 ⊢ ◡(𝐴 ∖ (𝐴 ∖ 𝐵)) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵))
5		dfin4 4198	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	5	cnveqi 5772	. 2 ⊢ ◡(𝐴 ∩ 𝐵) = ◡(𝐴 ∖ (𝐴 ∖ 𝐵))
7		dfin4 4198	. 2 ⊢ (◡𝐴 ∩ ◡𝐵) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵))
8	4, 6, 7	3eqtr4i 2776	1 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)

Syntax hints:   = wceq 1539   ∖ cdif 3880   ∩ cin 3882  ^◡ccnv 5579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588

This theorem is referenced by:  rnin  6039  dminxp  6072  imainrect  6073  cnvcnv  6084  cnvrescnv  6087  pjdm  20824  ordtrest2  22263  ustexsym  23275  trust  23289  ordtcnvNEW  31772  ordtrest2NEW  31775  msrf  33404  elrn3  33635  pprodcnveq  34112