Theorem cnvin 6129

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 6128	. . 3 ⊢ ◡(𝐴 ∖ (𝐴 ∖ 𝐵)) = (◡𝐴 ∖ ◡(𝐴 ∖ 𝐵))
2		cnvdif 6128	. . . 4 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵)
3	2	difeq2i 4070	. . 3 ⊢ (◡𝐴 ∖ ◡(𝐴 ∖ 𝐵)) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵))
4	1, 3	eqtri 2783	. 2 ⊢ ◡(𝐴 ∖ (𝐴 ∖ 𝐵)) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵))
5		dfin4 4223	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	5	cnveqi 5848	. 2 ⊢ ◡(𝐴 ∩ 𝐵) = ◡(𝐴 ∖ (𝐴 ∖ 𝐵))
7		dfin4 4223	. 2 ⊢ (◡𝐴 ∩ ◡𝐵) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵))
8	4, 6, 7	3eqtr4i 2793	1 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 3895   ∩ cin 3897  ^◡ccnv 5646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-cnv 5655

This theorem is used by:  rninOLD  6132  dminxp  6167  imainrect  6168  cnvcnv  6179  cnvrescnv  6183  pjdm  21974  ordtrest2  23483  ustexsym  24496  trust  24509  ordtcnvNEW  34485  ordtrest2NEW  34488  msrf  36228  elrn3  36448  pprodcnveq  36567  tposrescnv  49909