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Theorem dfif3 3455
Description: Alternate definition of the conditional operator df-if 3443. 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 3444 . 2  |-  if (
ph ,  A ,  B )  =  ( { y  e.  A  |  ph }  u.  {
y  e.  B  |  -.  ph } )
2 dfif3.1 . . . . . 6  |-  C  =  { x  |  ph }
3 biidd 171 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<-> 
ph ) )
43cbvabv 2239 . . . . . 6  |-  { x  |  ph }  =  {
y  |  ph }
52, 4eqtri 2136 . . . . 5  |-  C  =  { y  |  ph }
65ineq2i 3242 . . . 4  |-  ( A  i^i  C )  =  ( A  i^i  {
y  |  ph }
)
7 dfrab3 3320 . . . 4  |-  { y  e.  A  |  ph }  =  ( A  i^i  { y  |  ph } )
86, 7eqtr4i 2139 . . 3  |-  ( A  i^i  C )  =  { y  e.  A  |  ph }
9 dfrab3 3320 . . . 4  |-  { y  e.  B  |  -.  ph }  =  ( B  i^i  { y  |  -.  ph } )
10 notab 3314 . . . . . 6  |-  { y  |  -.  ph }  =  ( _V  \  { y  |  ph } )
115difeq2i 3159 . . . . . 6  |-  ( _V 
\  C )  =  ( _V  \  {
y  |  ph }
)
1210, 11eqtr4i 2139 . . . . 5  |-  { y  |  -.  ph }  =  ( _V  \  C )
1312ineq2i 3242 . . . 4  |-  ( B  i^i  { y  |  -.  ph } )  =  ( B  i^i  ( _V  \  C ) )
149, 13eqtr2i 2137 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  { y  e.  B  |  -.  ph }
158, 14uneq12i 3196 . 2  |-  ( ( A  i^i  C )  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( { y  e.  A  |  ph }  u.  {
y  e.  B  |  -.  ph } )
161, 15eqtr4i 2139 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 1314   {cab 2101   {crab 2395   _Vcvv 2658    \ cdif 3036    u. cun 3037    i^i cin 3038   ifcif 3442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-if 3443
This theorem is referenced by: (None)
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