Theorem dfif3 3343

Description: Alternate definition of the conditional operator df-if 3332. 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 3333	. 2 |- if ( ph , A , B ) = ( { y e. A | ph } u. { y e. B | -. ph } )
2		dfif3.1	. . . . . 6 |- C = { x | ph }
3		biidd 161	. . . . . . 7 |- ( x = y -> ( ph <-> ph ) )
4	3	cbvabv 2161	. . . . . 6 |- { x | ph } = { y | ph }
5	2, 4	eqtri 2060	. . . . 5 |- C = { y | ph }
6	5	ineq2i 3135	. . . 4 |- ( A i^i C ) = ( A i^i { y | ph } )
7		dfrab3 3213	. . . 4 |- { y e. A | ph } = ( A i^i { y | ph } )
8	6, 7	eqtr4i 2063	. . 3 |- ( A i^i C ) = { y e. A | ph }
9		dfrab3 3213	. . . 4 |- { y e. B | -. ph } = ( B i^i { y | -. ph } )
10		notab 3207	. . . . . 6 |- { y | -. ph } = ( _V \ { y | ph } )
11	5	difeq2i 3059	. . . . . 6 |- ( _V \ C ) = ( _V \ { y | ph } )
12	10, 11	eqtr4i 2063	. . . . 5 |- { y | -. ph } = ( _V \ C )
13	12	ineq2i 3135	. . . 4 |- ( B i^i { y | -. ph } ) = ( B i^i ( _V \ C ) )
14	9, 13	eqtr2i 2061	. . 3 |- ( B i^i ( _V \ C ) ) = { y e. B | -. ph }
15	8, 14	uneq12i 3095	. 2 |- ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) ) = ( { y e. A | ph } u. { y e. B | -. ph } )
16	1, 15	eqtr4i 2063	1 |- if ( ph , A , B ) = ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) )

Syntax hints:   -. wn 3   = wceq 1243   {cab 2026   {crab 2310   _Vcvv 2557   \ cdif 2914   u. cun 2915   i^i cin 2916   ifcif 3331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-if 3332

This theorem is referenced by: (None)