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Theorem dfif3 3371
Description: Alternate definition of the conditional operator df-if 3360. 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.
StepHypRef Expression
1 dfif6 3361 . 2  |-  if (
ph ,  A ,  B )  =  ( { y  e.  A  |  ph }  u.  {
y  e.  B  |  -.  ph } )
2 dfif3.1 . . . . . 6  |-  C  =  { x  |  ph }
3 biidd 165 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<-> 
ph ) )
43cbvabv 2177 . . . . . 6  |-  { x  |  ph }  =  {
y  |  ph }
52, 4eqtri 2076 . . . . 5  |-  C  =  { y  |  ph }
65ineq2i 3163 . . . 4  |-  ( A  i^i  C )  =  ( A  i^i  {
y  |  ph }
)
7 dfrab3 3241 . . . 4  |-  { y  e.  A  |  ph }  =  ( A  i^i  { y  |  ph } )
86, 7eqtr4i 2079 . . 3  |-  ( A  i^i  C )  =  { y  e.  A  |  ph }
9 dfrab3 3241 . . . 4  |-  { y  e.  B  |  -.  ph }  =  ( B  i^i  { y  |  -.  ph } )
10 notab 3235 . . . . . 6  |-  { y  |  -.  ph }  =  ( _V  \  { y  |  ph } )
115difeq2i 3087 . . . . . 6  |-  ( _V 
\  C )  =  ( _V  \  {
y  |  ph }
)
1210, 11eqtr4i 2079 . . . . 5  |-  { y  |  -.  ph }  =  ( _V  \  C )
1312ineq2i 3163 . . . 4  |-  ( B  i^i  { y  |  -.  ph } )  =  ( B  i^i  ( _V  \  C ) )
149, 13eqtr2i 2077 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  { y  e.  B  |  -.  ph }
158, 14uneq12i 3123 . 2  |-  ( ( A  i^i  C )  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( { y  e.  A  |  ph }  u.  {
y  e.  B  |  -.  ph } )
161, 15eqtr4i 2079 1  |-  if (
ph ,  A ,  B )  =  ( ( A  i^i  C
)  u.  ( B  i^i  ( _V  \  C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1259   {cab 2042   {crab 2327   _Vcvv 2574    \ cdif 2942    u. cun 2943    i^i cin 2944   ifcif 3359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-if 3360
This theorem is referenced by: (None)
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