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Theorem ifle 10783
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 9593 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  A  <_  A )
3 iftrue 3745 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
43adantl 453 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ph ,  A ,  B )  =  A )
5 id 20 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
65imp 419 . . . . 5  |-  ( ( ( ph  ->  ps )  /\  ph )  ->  ps )
76adantll 695 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  ps )
8 iftrue 3745 . . . 4  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
97, 8syl 16 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ps ,  A ,  B )  =  A )
102, 4, 93brtr4d 4242 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B ) )
11 iffalse 3746 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
1211adantl 453 . . 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 9593 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  B )
16 breq2 4216 . . . . 5  |-  ( A  =  if ( ps ,  A ,  B
)  ->  ( B  <_  A  <->  B  <_  if ( ps ,  A ,  B ) ) )
17 breq2 4216 . . . . 5  |-  ( B  =  if ( ps ,  A ,  B
)  ->  ( B  <_  B  <->  B  <_  if ( ps ,  A ,  B ) ) )
1816, 17ifboth 3770 . . . 4  |-  ( ( B  <_  A  /\  B  <_  B )  ->  B  <_  if ( ps ,  A ,  B
) )
1913, 15, 18syl2anc 643 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  if ( ps ,  A ,  B ) )
2012, 19eqbrtrd 4232 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B )
)
2110, 20pm2.61dan 767 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 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ifcif 3739   class class class wbr 4212   RRcr 8989    <_ cle 9121
This theorem is referenced by:  rpnnen2lem4  12817  itg2cnlem2  19654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-pre-lttri 9064
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126
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