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Theorem ifbi 3805
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 888 . 2  |-  ( (
ph 
<->  ps )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) ) )
2 iftrue 3792 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
3 iftrue 3792 . . . . 5  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
43eqcomd 2443 . . . 4  |-  ( ps 
->  A  =  if ( ps ,  A ,  B ) )
52, 4sylan9eq 2490 . . 3  |-  ( (
ph  /\  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B )
)
6 iffalse 3794 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
7 iffalse 3794 . . . . 5  |-  ( -. 
ps  ->  if ( ps ,  A ,  B
)  =  B )
87eqcomd 2443 . . . 4  |-  ( -. 
ps  ->  B  =  if ( ps ,  A ,  B ) )
96, 8sylan9eq 2490 . . 3  |-  ( ( -.  ph  /\  -.  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B
) )
105, 9jaoi 379 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  -.  ps )
)  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
111, 10sylbi 195 1  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369   ifcif 3786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-if 3787
This theorem is referenced by:  ifbid  3806  ifbieq2i  3808  mulmarep1gsum1  18359  madugsum  18424  dchrhash  22585  lgsdi  22646  rpvmasum2  22736  itg2gt0cn  28400  bj-projval  32336  elimhyps  32452  dedths  32453
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