Theorem dfif3 3364

Description: Alternate definition of the conditional operator df-if 3352. Note that ph is independent of x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)

Hypothesis

Ref	Expression
dfif3.1	|- C = { x | ph }

Assertion

Ref	Expression
dfif3	|- if ( ph , A , B ) = ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) )

Distinct variable group:   ph, x

Allowed substitution hints:   A( x)   B( x)   C( x)

Proof of Theorem dfif3

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfif6 3353	. 2 |- if ( ph , A , B ) = ( { y e. A | ph } u. { y e. B | -. ph } )
2		dfif3.1	. . . . . 6 |- C = { x | ph }
3		biidd 170	. . . . . . 7 |- ( x = y -> ( ph <-> ph ) )
4	3	cbvabv 2202	. . . . . 6 |- { x | ph } = { y | ph }
5	2, 4	eqtri 2101	. . . . 5 |- C = { y | ph }
6	5	ineq2i 3164	. . . 4 |- ( A i^i C ) = ( A i^i { y | ph } )
7		dfrab3 3240	. . . 4 |- { y e. A | ph } = ( A i^i { y | ph } )
8	6, 7	eqtr4i 2104	. . 3 |- ( A i^i C ) = { y e. A | ph }
9		dfrab3 3240	. . . 4 |- { y e. B | -. ph } = ( B i^i { y | -. ph } )
10		notab 3234	. . . . . 6 |- { y | -. ph } = ( _V \ { y | ph } )
11	5	difeq2i 3087	. . . . . 6 |- ( _V \ C ) = ( _V \ { y | ph } )
12	10, 11	eqtr4i 2104	. . . . 5 |- { y | -. ph } = ( _V \ C )
13	12	ineq2i 3164	. . . 4 |- ( B i^i { y | -. ph } ) = ( B i^i ( _V \ C ) )
14	9, 13	eqtr2i 2102	. . 3 |- ( B i^i ( _V \ C ) ) = { y e. B | -. ph }
15	8, 14	uneq12i 3124	. 2 |- ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) ) = ( { y e. A | ph } u. { y e. B | -. ph } )
16	1, 15	eqtr4i 2104	1 |- if ( ph , A , B ) = ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) )

Syntax hints:   -. wn 3   = wceq 1284   {cab 2067   {crab 2352   _Vcvv 2601   \ cdif 2970   u. cun 2971   i^i cin 2972   ifcif 3351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-if 3352

This theorem is referenced by: (None)