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Theorem ifbi 3577
Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
Assertion
Ref Expression
ifbi  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )

Proof of Theorem ifbi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 anbi2 467 . . . 4  |-  ( (
ph 
<->  ps )  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
) )
2 notbi 667 . . . . 5  |-  ( (
ph 
<->  ps )  ->  ( -.  ph  <->  -.  ps )
)
32anbi2d 464 . . . 4  |-  ( (
ph 
<->  ps )  ->  (
( x  e.  B  /\  -.  ph )  <->  ( x  e.  B  /\  -.  ps ) ) )
41, 3orbi12d 794 . . 3  |-  ( (
ph 
<->  ps )  ->  (
( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) )  <-> 
( ( x  e.  A  /\  ps )  \/  ( x  e.  B  /\  -.  ps ) ) ) )
54abbidv 2311 . 2  |-  ( (
ph 
<->  ps )  ->  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }  =  { x  |  (
( x  e.  A  /\  ps )  \/  (
x  e.  B  /\  -.  ps ) ) } )
6 df-if 3558 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
7 df-if 3558 . 2  |-  if ( ps ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ps )  \/  (
x  e.  B  /\  -.  ps ) ) }
85, 6, 73eqtr4g 2251 1  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2164   {cab 2179   ifcif 3557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-if 3558
This theorem is referenced by:  ifbid  3578  ifbieq2i  3580  ifnebibdc  3600  fodjuomni  7208  fodjumkv  7219  nninfwlpoimlemg  7234  1tonninf  10512  lgsdi  15153  nninfsellemqall  15505  nninfomni  15509
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