Theorem dfin3 3866

Description: Intersection defined in terms of union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)

Assertion

Ref	Expression
dfin3	|- ( A i^i B ) = ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) )

Proof of Theorem dfin3

Step	Hyp	Ref	Expression
1		ddif 3742	. 2 |- ( _V \ ( _V \ ( A \ ( _V \ B ) ) ) ) = ( A \ ( _V \ B ) )
2		dfun2 3859	. . . 4 |- ( ( _V \ A ) u. ( _V \ B ) ) = ( _V \ ( ( _V \ ( _V \ A ) ) \ ( _V \ B ) ) )
3		ddif 3742	. . . . . 6 |- ( _V \ ( _V \ A ) ) = A
4	3	difeq1i 3724	. . . . 5 |- ( ( _V \ ( _V \ A ) ) \ ( _V \ B ) ) = ( A \ ( _V \ B ) )
5	4	difeq2i 3725	. . . 4 |- ( _V \ ( ( _V \ ( _V \ A ) ) \ ( _V \ B ) ) ) = ( _V \ ( A \ ( _V \ B ) ) )
6	2, 5	eqtri 2644	. . 3 |- ( ( _V \ A ) u. ( _V \ B ) ) = ( _V \ ( A \ ( _V \ B ) ) )
7	6	difeq2i 3725	. 2 |- ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) = ( _V \ ( _V \ ( A \ ( _V \ B ) ) ) )
8		dfin2 3860	. 2 |- ( A i^i B ) = ( A \ ( _V \ B ) )
9	1, 7, 8	3eqtr4ri 2655	1 |- ( A i^i B ) = ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) )

Syntax hints:   = wceq 1483   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581

This theorem is referenced by:  difindi  3881