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Theorem ifbi 3584
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
StepHypRef Expression
1 dfbi3 863 . 2  |-  ( (
ph 
<->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )
2 iftrue 3573 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
3 iftrue 3573 . . . . 5  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
43eqcomd 2290 . . . 4  |-  ( ps 
->  A  =  if ( ps ,  A ,  B ) )
52, 4sylan9eq 2337 . . 3  |-  ( (
ph  /\  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B )
)
6 iffalse 3574 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
7 iffalse 3574 . . . . 5  |-  ( -. 
ps  ->  if ( ps ,  A ,  B
)  =  B )
87eqcomd 2290 . . . 4  |-  ( -. 
ps  ->  B  =  if ( ps ,  A ,  B ) )
96, 8sylan9eq 2337 . . 3  |-  ( ( -.  ph  /\  -.  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B
) )
105, 9jaoi 368 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  -.  ps )
)  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
111, 10sylbi 187 1  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1625   ifcif 3567
This theorem is referenced by:  ifbid  3585  ifbieq2i  3586  dchrhash  20512  lgsdi  20573  rpvmasum2  20663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-if 3568
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