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Theorem dfif4 3752
Description: Alternate definition of the conditional operator df-if 3742. Note that  ph is independent of  x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
Hypothesis
Ref Expression
dfif3.1  |-  C  =  { x  |  ph }
Assertion
Ref Expression
dfif4  |-  if (
ph ,  A ,  B )  =  ( ( A  u.  B
)  i^i  ( ( A  u.  ( _V  \  C ) )  i^i  ( B  u.  C
) ) )
Distinct variable group:    ph, x
Allowed substitution hints:    A( x)    B( x)    C( x)

Proof of Theorem dfif4
StepHypRef Expression
1 dfif3.1 . . 3  |-  C  =  { x  |  ph }
21dfif3 3751 . 2  |-  if (
ph ,  A ,  B )  =  ( ( A  i^i  C
)  u.  ( B  i^i  ( _V  \  C ) ) )
3 undir 3592 . 2  |-  ( ( A  i^i  C )  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  u.  ( B  i^i  ( _V  \  C ) ) )  i^i  ( C  u.  ( B  i^i  ( _V  \  C ) ) ) )
4 undi 3590 . . . 4  |-  ( A  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  u.  B
)  i^i  ( A  u.  ( _V  \  C
) ) )
5 undi 3590 . . . . 5  |-  ( C  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( ( C  u.  B
)  i^i  ( C  u.  ( _V  \  C
) ) )
6 uncom 3493 . . . . . 6  |-  ( C  u.  B )  =  ( B  u.  C
)
7 undifv 3704 . . . . . 6  |-  ( C  u.  ( _V  \  C ) )  =  _V
86, 7ineq12i 3542 . . . . 5  |-  ( ( C  u.  B )  i^i  ( C  u.  ( _V  \  C ) ) )  =  ( ( B  u.  C
)  i^i  _V )
9 inv1 3656 . . . . 5  |-  ( ( B  u.  C )  i^i  _V )  =  ( B  u.  C
)
105, 8, 93eqtri 2462 . . . 4  |-  ( C  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( B  u.  C )
114, 10ineq12i 3542 . . 3  |-  ( ( A  u.  ( B  i^i  ( _V  \  C ) ) )  i^i  ( C  u.  ( B  i^i  ( _V  \  C ) ) ) )  =  ( ( ( A  u.  B )  i^i  ( A  u.  ( _V  \  C ) ) )  i^i  ( B  u.  C ) )
12 inass 3553 . . 3  |-  ( ( ( A  u.  B
)  i^i  ( A  u.  ( _V  \  C
) ) )  i^i  ( B  u.  C
) )  =  ( ( A  u.  B
)  i^i  ( ( A  u.  ( _V  \  C ) )  i^i  ( B  u.  C
) ) )
1311, 12eqtri 2458 . 2  |-  ( ( A  u.  ( B  i^i  ( _V  \  C ) ) )  i^i  ( C  u.  ( B  i^i  ( _V  \  C ) ) ) )  =  ( ( A  u.  B
)  i^i  ( ( A  u.  ( _V  \  C ) )  i^i  ( B  u.  C
) ) )
142, 3, 133eqtri 2462 1  |-  if (
ph ,  A ,  B )  =  ( ( A  u.  B
)  i^i  ( ( A  u.  ( _V  \  C ) )  i^i  ( B  u.  C
) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653   {cab 2424   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321   ifcif 3741
This theorem is referenced by:  dfif5  3753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742
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