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Theorem dfif3 3575
Description: Alternate definition of the conditional operator df-if 3566. 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 3568 . 2  |-  if (
ph ,  A ,  B )  =  ( { y  e.  A  |  ph }  u.  {
y  e.  B  |  -.  ph } )
2 dfif3.1 . . . . . 6  |-  C  =  { x  |  ph }
3 biidd 228 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<-> 
ph ) )
43cbvabv 2402 . . . . . 6  |-  { x  |  ph }  =  {
y  |  ph }
52, 4eqtri 2303 . . . . 5  |-  C  =  { y  |  ph }
65ineq2i 3367 . . . 4  |-  ( A  i^i  C )  =  ( A  i^i  {
y  |  ph }
)
7 dfrab3 3444 . . . 4  |-  { y  e.  A  |  ph }  =  ( A  i^i  { y  |  ph } )
86, 7eqtr4i 2306 . . 3  |-  ( A  i^i  C )  =  { y  e.  A  |  ph }
9 dfrab3 3444 . . . 4  |-  { y  e.  B  |  -.  ph }  =  ( B  i^i  { y  |  -.  ph } )
10 notab 3438 . . . . . 6  |-  { y  |  -.  ph }  =  ( _V  \  { y  |  ph } )
115difeq2i 3291 . . . . . 6  |-  ( _V 
\  C )  =  ( _V  \  {
y  |  ph }
)
1210, 11eqtr4i 2306 . . . . 5  |-  { y  |  -.  ph }  =  ( _V  \  C )
1312ineq2i 3367 . . . 4  |-  ( B  i^i  { y  |  -.  ph } )  =  ( B  i^i  ( _V  \  C ) )
149, 13eqtr2i 2304 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  { y  e.  B  |  -.  ph }
158, 14uneq12i 3327 . 2  |-  ( ( A  i^i  C )  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( { y  e.  A  |  ph }  u.  {
y  e.  B  |  -.  ph } )
161, 15eqtr4i 2306 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 1623   {cab 2269   {crab 2547   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151   ifcif 3565
This theorem is referenced by:  dfif4  3576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-if 3566
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