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Theorem dfin2 4261
Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 4260. Another version is given by dfin4 4268. (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 3475 . . . . . 6 𝑥 ∈ V
2 eldif 3957 . . . . . 6 (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐵))
31, 2mpbiran 708 . . . . 5 (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥𝐵)
43con2bii 357 . . . 4 (𝑥𝐵 ↔ ¬ 𝑥 ∈ (V ∖ 𝐵))
54anbi2i 622 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
6 eldif 3957 . . 3 (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
75, 6bitr4i 278 . 2 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
87ineqri 4204 1 (𝐴𝐵) = (𝐴 ∖ (V ∖ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1534  wcel 2099  Vcvv 3471  cdif 3944  cin 3946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-dif 3950  df-in 3954
This theorem is referenced by:  dfun3  4266  dfin3  4267  invdif  4269  difundi  4280  difindi  4282  difdif2  4287
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