MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfif3 Structured version   Unicode version

Theorem dfif3 3914
Description: Alternate definition of the conditional operator df-if 3903. 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 3905 . 2  |-  if (
ph ,  A ,  B )  =  ( { y  e.  A  |  ph }  u.  {
y  e.  B  |  -.  ph } )
2 dfif3.1 . . . . . 6  |-  C  =  { x  |  ph }
3 biidd 237 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<-> 
ph ) )
43cbvabv 2597 . . . . . 6  |-  { x  |  ph }  =  {
y  |  ph }
52, 4eqtri 2483 . . . . 5  |-  C  =  { y  |  ph }
65ineq2i 3660 . . . 4  |-  ( A  i^i  C )  =  ( A  i^i  {
y  |  ph }
)
7 dfrab3 3736 . . . 4  |-  { y  e.  A  |  ph }  =  ( A  i^i  { y  |  ph } )
86, 7eqtr4i 2486 . . 3  |-  ( A  i^i  C )  =  { y  e.  A  |  ph }
9 dfrab3 3736 . . . 4  |-  { y  e.  B  |  -.  ph }  =  ( B  i^i  { y  |  -.  ph } )
10 notab 3731 . . . . . 6  |-  { y  |  -.  ph }  =  ( _V  \  { y  |  ph } )
115difeq2i 3582 . . . . . 6  |-  ( _V 
\  C )  =  ( _V  \  {
y  |  ph }
)
1210, 11eqtr4i 2486 . . . . 5  |-  { y  |  -.  ph }  =  ( _V  \  C )
1312ineq2i 3660 . . . 4  |-  ( B  i^i  { y  |  -.  ph } )  =  ( B  i^i  ( _V  \  C ) )
149, 13eqtr2i 2484 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  { y  e.  B  |  -.  ph }
158, 14uneq12i 3619 . 2  |-  ( ( A  i^i  C )  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( { y  e.  A  |  ph }  u.  {
y  e.  B  |  -.  ph } )
161, 15eqtr4i 2486 1  |-  if (
ph ,  A ,  B )  =  ( ( A  i^i  C
)  u.  ( B  i^i  ( _V  \  C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370   {cab 2439   {crab 2803   _Vcvv 3078    \ cdif 3436    u. cun 3437    i^i cin 3438   ifcif 3902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-if 3903
This theorem is referenced by:  dfif4  3915
  Copyright terms: Public domain W3C validator