MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ifbi Unicode version

Theorem ifbi 3595
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 3584 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
3 iftrue 3584 . . . . 5  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
43eqcomd 2301 . . . 4  |-  ( ps 
->  A  =  if ( ps ,  A ,  B ) )
52, 4sylan9eq 2348 . . 3  |-  ( (
ph  /\  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B )
)
6 iffalse 3585 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
7 iffalse 3585 . . . . 5  |-  ( -. 
ps  ->  if ( ps ,  A ,  B
)  =  B )
87eqcomd 2301 . . . 4  |-  ( -. 
ps  ->  B  =  if ( ps ,  A ,  B ) )
96, 8sylan9eq 2348 . . 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 1632   ifcif 3578
This theorem is referenced by:  ifbid  3596  ifbieq2i  3597  dchrhash  20526  lgsdi  20587  rpvmasum2  20677  itg2gt0cn  25006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-if 3579
  Copyright terms: Public domain W3C validator