Theorem dfif3 4100

Description: Alternate definition of the conditional operator df-if 4087. 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 4089	. 2 |- if ( ph , A , B ) = ( { y e. A | ph } u. { y e. B | -. ph } )
2		dfif3.1	. . . . . 6 |- C = { x | ph }
3		biidd 252	. . . . . . 7 |- ( x = y -> ( ph <-> ph ) )
4	3	cbvabv 2747	. . . . . 6 |- { x | ph } = { y | ph }
5	2, 4	eqtri 2644	. . . . 5 |- C = { y | ph }
6	5	ineq2i 3811	. . . 4 |- ( A i^i C ) = ( A i^i { y | ph } )
7		dfrab3 3902	. . . 4 |- { y e. A | ph } = ( A i^i { y | ph } )
8	6, 7	eqtr4i 2647	. . 3 |- ( A i^i C ) = { y e. A | ph }
9		dfrab3 3902	. . . 4 |- { y e. B | -. ph } = ( B i^i { y | -. ph } )
10		notab 3897	. . . . . 6 |- { y | -. ph } = ( _V \ { y | ph } )
11	5	difeq2i 3725	. . . . . 6 |- ( _V \ C ) = ( _V \ { y | ph } )
12	10, 11	eqtr4i 2647	. . . . 5 |- { y | -. ph } = ( _V \ C )
13	12	ineq2i 3811	. . . 4 |- ( B i^i { y | -. ph } ) = ( B i^i ( _V \ C ) )
14	9, 13	eqtr2i 2645	. . 3 |- ( B i^i ( _V \ C ) ) = { y e. B | -. ph }
15	8, 14	uneq12i 3765	. 2 |- ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) ) = ( { y e. A | ph } u. { y e. B | -. ph } )
16	1, 15	eqtr4i 2647	1 |- if ( ph , A , B ) = ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) )

Syntax hints:   -. wn 3   = wceq 1483   {cab 2608   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   ifcif 4086

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  df-if 4087

This theorem is referenced by:  dfif4  4101