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Theorem dfin2 4225
Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 4224. Another version is given by dfin4 4232. (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 3452 . . . . . 6 𝑥 ∈ V
2 eldif 3925 . . . . . 6 (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐵))
31, 2mpbiran 708 . . . . 5 (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥𝐵)
43con2bii 358 . . . 4 (𝑥𝐵 ↔ ¬ 𝑥 ∈ (V ∖ 𝐵))
54anbi2i 624 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
6 eldif 3925 . . 3 (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
75, 6bitr4i 278 . 2 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
87ineqri 4169 1 (𝐴𝐵) = (𝐴 ∖ (V ∖ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397   = wceq 1542  wcel 2107  Vcvv 3448  cdif 3912  cin 3914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-dif 3918  df-in 3922
This theorem is referenced by:  dfun3  4230  dfin3  4231  invdif  4233  difundi  4244  difindi  4246  difdif2  4251
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