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Theorem ifbi 3927
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 901 . 2  |-  ( (
ph 
<->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )
2 iftrue 3912 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
3 iftrue 3912 . . . . 5  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
43eqcomd 2428 . . . 4  |-  ( ps 
->  A  =  if ( ps ,  A ,  B ) )
52, 4sylan9eq 2481 . . 3  |-  ( (
ph  /\  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B )
)
6 iffalse 3915 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
7 iffalse 3915 . . . . 5  |-  ( -. 
ps  ->  if ( ps ,  A ,  B
)  =  B )
87eqcomd 2428 . . . 4  |-  ( -. 
ps  ->  B  =  if ( ps ,  A ,  B ) )
96, 8sylan9eq 2481 . . 3  |-  ( ( -.  ph  /\  -.  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B
) )
105, 9jaoi 380 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  -.  ps )
)  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
111, 10sylbi 198 1  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437   ifcif 3906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-if 3907
This theorem is referenced by:  ifbid  3928  ifbieq2i  3930  gsummoncoe1  18826  scmatscm  19462  mulmarep1gsum1  19522  madugsum  19592  mp2pm2mplem4  19757  dchrhash  24088  lgsdi  24149  rpvmasum2  24239  bj-projval  31365  itg2gt0cn  31730  elimhyps  32271  dedths  32272
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