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Theorem ifle 10490
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 999 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  A  e.  RR )
21leidd 9307 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  A  <_  A )
3 iftrue 3545 . . . 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 697 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  ps )
8 iftrue 3545 . . . 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 4027 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  ph )  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B ) )
11 iffalse 3546 . . . 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 1001 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  A )
14 simpll2 1000 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  e.  RR )
1514leidd 9307 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  B )
16 breq2 4001 . . . . 5  |-  ( A  =  if ( ps ,  A ,  B
)  ->  ( B  <_  A  <->  B  <_  if ( ps ,  A ,  B ) ) )
17 breq2 4001 . . . . 5  |-  ( B  =  if ( ps ,  A ,  B
)  ->  ( B  <_  B  <->  B  <_  if ( ps ,  A ,  B ) ) )
1816, 17ifboth 3570 . . . 4  |-  ( ( B  <_  A  /\  B  <_  B )  ->  B  <_  if ( ps ,  A ,  B
) )
1913, 15, 18syl2anc 645 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  B  <_  if ( ps ,  A ,  B ) )
2012, 19eqbrtrd 4017 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  B  <_  A )  /\  ( ph  ->  ps ) )  /\  -.  ph )  ->  if ( ph ,  A ,  B )  <_  if ( ps ,  A ,  B )
)
2110, 20pm2.61dan 769 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 939    = wceq 1619    e. wcel 1621   ifcif 3539   class class class wbr 3997   RRcr 8704    <_ cle 8836
This theorem is referenced by:  rpnnen2lem4  12458  itg2cnlem2  19079
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-pre-lttri 8779
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841
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