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Theorem bj-df-ifc 30716
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 2388. (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 30715 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( (
ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }
2 df-ifp 1387 . . . 4  |-  (if- (
ph ,  x  e.  A ,  x  e.  B )  <->  ( ( ph  /\  x  e.  A
)  \/  ( -. 
ph  /\  x  e.  B ) ) )
32bicomi 202 . . 3  |-  ( ( ( ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) )  <-> if- ( ph ,  x  e.  A ,  x  e.  B )
)
43abbii 2536 . 2  |-  { x  |  ( ( ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }  =  {
x  | if- ( ph ,  x  e.  A ,  x  e.  B
) }
51, 4eqtri 2431 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 366    /\ wa 367  if-wif 1386    = wceq 1405    e. wcel 1842   {cab 2387   ifcif 3884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ifp 1387  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-if 3885
This theorem is referenced by:  bj-ififc  30717
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