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Theorem bj-df-ifc 33123
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 2448. (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 33122 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( (
ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }
2 bj-dfif5 33106 . . . 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 2596 . 2  |-  { x  |  ( ( ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }  =  {
x  | if- ( ph ,  x  e.  A ,  x  e.  B
) }
51, 4eqtri 2491 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 368    /\ wa 369    = wceq 1374    e. wcel 1762   {cab 2447   ifcif 3934  if-wif 33101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-if 3935  df-bj-if 33102
This theorem is referenced by:  bj-ififc  33124
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