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Theorem bj-df-ifc 31159
Description: The definition of "ifc" if "if-" enters the main part. This is in line with the definition of a class as the extension of a predicate in df-clab 2438. (Contributed by BJ, 20-Sep-2019.)
Assertion
Ref Expression
bj-df-ifc  |-  if (
ph ,  A ,  B )  =  {
x  | if- ( ph ,  x  e.  A ,  x  e.  B
) }
Distinct variable groups:    ph, x    x, A    x, B

Proof of Theorem bj-df-ifc
StepHypRef Expression
1 bj-dfifc2 31158 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( (
ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }
2 df-ifp 1426 . . . 4  |-  (if- (
ph ,  x  e.  A ,  x  e.  B )  <->  ( ( ph  /\  x  e.  A
)  \/  ( -. 
ph  /\  x  e.  B ) ) )
32bicomi 206 . . 3  |-  ( ( ( ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) )  <-> if- ( ph ,  x  e.  A ,  x  e.  B )
)
43abbii 2567 . 2  |-  { x  |  ( ( ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }  =  {
x  | if- ( ph ,  x  e.  A ,  x  e.  B
) }
51, 4eqtri 2473 1  |-  if (
ph ,  A ,  B )  =  {
x  | if- ( ph ,  x  e.  A ,  x  e.  B
) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 370    /\ wa 371  if-wif 1425    = wceq 1444    e. wcel 1887   {cab 2437   ifcif 3881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-ifp 1426  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-if 3882
This theorem is referenced by:  bj-ififc  31160
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