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Theorem bj-dfifc2 33120
Description: This should be the alternate definition of "ifc" if "if-" enters the main part. (Contributed by BJ, 20-Sep-2019.)
Assertion
Ref Expression
bj-dfifc2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( (
ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }
Distinct variable groups:    ph, x    x, A    x, B

Proof of Theorem bj-dfifc2
StepHypRef Expression
1 df-if 3933 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
2 ancom 450 . . . . 5  |-  ( (
ph  /\  x  e.  A )  <->  ( x  e.  A  /\  ph )
)
3 ancom 450 . . . . 5  |-  ( ( -.  ph  /\  x  e.  B )  <->  ( x  e.  B  /\  -.  ph ) )
42, 3orbi12i 521 . . . 4  |-  ( ( ( ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) )  <->  ( (
x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) )
54bicomi 202 . . 3  |-  ( ( ( x  e.  A  /\  ph )  \/  (
x  e.  B  /\  -.  ph ) )  <->  ( ( ph  /\  x  e.  A
)  \/  ( -. 
ph  /\  x  e.  B ) ) )
65abbii 2594 . 2  |-  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }  =  { x  |  (
( ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }
71, 6eqtri 2489 1  |-  if (
ph ,  A ,  B )  =  {
x  |  ( (
ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762   {cab 2445   ifcif 3932
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 1961  ax-ext 2438
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 2446  df-cleq 2452  df-clel 2455  df-if 3933
This theorem is referenced by:  bj-df-ifc  33121
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