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Theorem dfin2 3844
Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3843. Another version is given by dfin4 3849. (Contributed by NM, 10-Jun-2004.)
Assertion
Ref Expression
dfin2 (𝐴𝐵) = (𝐴 ∖ (V ∖ 𝐵))

Proof of Theorem dfin2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3193 . . . . . 6 𝑥 ∈ V
2 eldif 3570 . . . . . 6 (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐵))
31, 2mpbiran 952 . . . . 5 (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥𝐵)
43con2bii 347 . . . 4 (𝑥𝐵 ↔ ¬ 𝑥 ∈ (V ∖ 𝐵))
54anbi2i 729 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
6 eldif 3570 . . 3 (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
75, 6bitr4i 267 . 2 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
87ineqri 3790 1 (𝐴𝐵) = (𝐴 ∖ (V ∖ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  cdif 3557  cin 3559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-dif 3563  df-in 3567
This theorem is referenced by:  dfun3  3847  dfin3  3848  invdif  3850  difundi  3861  difindi  3863  difdif2  3866
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