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Theorem ifle 10520
Description: An if statement transforms an implication into an inequality of terms. (Contributed by Mario Carneiro, 31-Aug-2014.)
Assertion
Ref Expression
ifle  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps )
)  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B ) )

Proof of Theorem ifle
StepHypRef Expression
1 simpll1 996 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  A  e.  RR )
21leidd 9336 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  A  <_  A )
3 iftrue 3574 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
43adantl 454 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ph ,  A ,  B )  =  A )
5 id 21 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
65imp 420 . . . . 5  |-  ( ( ( ph  ->  ps )  /\  ph )  ->  ps )
76adantll 696 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  ps )
8 iftrue 3574 . . . 4  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
97, 8syl 17 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ps ,  A ,  B )  =  A )
102, 4, 93brtr4d 4056 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B ) )
11 iffalse 3575 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
1211adantl 454 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  if ( ph ,  A ,  B )  =  B )
13 simpll3 998 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  A )
14 simpll2 997 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  e.  RR )
1514leidd 9336 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  B )
16 breq2 4030 . . . . 5  |-  ( A  =  if ( ps ,  A ,  B
)  ->  ( B  <_  A  <->  B  <_  if ( ps ,  A ,  B ) ) )
17 breq2 4030 . . . . 5  |-  ( B  =  if ( ps ,  A ,  B
)  ->  ( B  <_  B  <->  B  <_  if ( ps ,  A ,  B ) ) )
1816, 17ifboth 3599 . . . 4  |-  ( ( B  <_  A  /\  B  <_  B )  ->  B  <_  if ( ps ,  A ,  B
) )
1913, 15, 18syl2anc 644 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  if ( ps ,  A ,  B ) )
2012, 19eqbrtrd 4046 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B )
)
2110, 20pm2.61dan 768 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps )
)  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1625    e. wcel 1687   ifcif 3568   class class class wbr 4026   RRcr 8733    <_ cle 8865
This theorem is referenced by:  rpnnen2lem4  12492  itg2cnlem2  19113
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-resscn 8791  ax-pre-lttri 8808
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-br 4027  df-opab 4081  df-mpt 4082  df-id 4310  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-er 6657  df-en 6861  df-dom 6862  df-sdom 6863  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870
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