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Theorem dfif4 3924
Description: Alternate definition of the conditional operator df-if 3910. 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 3923 . 2  |-  if (
ph ,  A ,  B )  =  ( ( A  i^i  C
)  u.  ( B  i^i  ( _V  \  C ) ) )
3 undir 3722 . 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 3720 . . . 4  |-  ( A  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  u.  B
)  i^i  ( A  u.  ( _V  \  C
) ) )
5 undi 3720 . . . . 5  |-  ( C  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( ( C  u.  B
)  i^i  ( C  u.  ( _V  \  C
) ) )
6 uncom 3610 . . . . . 6  |-  ( C  u.  B )  =  ( B  u.  C
)
7 unvdif 3869 . . . . . 6  |-  ( C  u.  ( _V  \  C ) )  =  _V
86, 7ineq12i 3662 . . . . 5  |-  ( ( C  u.  B )  i^i  ( C  u.  ( _V  \  C ) ) )  =  ( ( B  u.  C
)  i^i  _V )
9 inv1 3789 . . . . 5  |-  ( ( B  u.  C )  i^i  _V )  =  ( B  u.  C
)
105, 8, 93eqtri 2455 . . . 4  |-  ( C  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( B  u.  C )
114, 10ineq12i 3662 . . 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 3672 . . 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 2451 . 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 2455 1  |-  if (
ph ,  A ,  B )  =  ( ( A  u.  B
)  i^i  ( ( A  u.  ( _V  \  C ) )  i^i  ( B  u.  C
) ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   {cab 2407   _Vcvv 3081    \ cdif 3433    u. cun 3434    i^i cin 3435   ifcif 3909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910
This theorem is referenced by:  dfif5  3925
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