ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  genpdflem Structured version   Unicode version

Theorem genpdflem 6490
Description: Simplification of upper or lower cut expression. Lemma for genpdf 6491. (Contributed by Jim Kingdon, 30-Sep-2019.)
Hypotheses
Ref Expression
genpdflem.r  r  r  Q.
genpdflem.s  s  s  Q.
Assertion
Ref Expression
genpdflem  { q  Q.  |  r  Q.  s  Q. 
r  s  q  r G s }  { q  Q.  |  r  s  q  r G s }
Distinct variable groups:   , s   , q,
r, s
Allowed substitution hints:   ( r, q)   ( s, r, q)    G( s, r, q)

Proof of Theorem genpdflem
StepHypRef Expression
1 genpdflem.r . . . . . . . . 9  r  r  Q.
21ex 108 . . . . . . . 8  r  r  Q.
32pm4.71rd 374 . . . . . . 7  r  r  Q.  r
43anbi1d 438 . . . . . 6  r  s  Q.  s  q  r G s  r  Q.  r  s  Q.  s  q 
r G s
54exbidv 1703 . . . . 5  r r  s  Q. 
s  q  r G s  r r  Q.  r  s  Q.  s  q 
r G s
6 3anass 888 . . . . . . . . . 10  r  s  q  r G s  r  s  q 
r G s
76rexbii 2325 . . . . . . . . 9  s  Q. 
r  s  q  r G s  s  Q.  r  s  q 
r G s
8 r19.42v 2461 . . . . . . . . 9  s  Q. 
r  s  q 
r G s  r  s  Q. 
s  q  r G s
97, 8bitri 173 . . . . . . . 8  s  Q. 
r  s  q  r G s  r  s  Q.  s  q 
r G s
109rexbii 2325 . . . . . . 7  r  Q.  s  Q.  r  s  q  r G s  r  Q.  r  s  Q.  s  q  r G s
11 df-rex 2306 . . . . . . 7  r  Q. 
r  s  Q. 
s  q  r G s  r r  Q.  r  s  Q.  s  q 
r G s
1210, 11bitri 173 . . . . . 6  r  Q.  s  Q.  r  s  q  r G s  r r  Q.  r  s  Q.  s  q 
r G s
13 anass 381 . . . . . . 7  r  Q.  r  s  Q.  s  q 
r G s  r  Q. 
r  s  Q. 
s  q  r G s
1413exbii 1493 . . . . . 6  r r  Q.  r  s  Q. 
s  q  r G s  r r  Q.  r  s  Q.  s  q 
r G s
1512, 14bitr4i 176 . . . . 5  r  Q.  s  Q.  r  s  q  r G s  r r  Q.  r  s  Q.  s  q 
r G s
165, 15syl6rbbr 188 . . . 4  r 
Q.  s  Q.  r  s  q 
r G s  r r  s  Q. 
s  q  r G s
17 df-rex 2306 . . . 4  r  s  Q.  s  q  r G s  r r  s  Q.  s  q 
r G s
1816, 17syl6bbr 187 . . 3  r 
Q.  s  Q.  r  s  q 
r G s  r  s 
Q.  s  q  r G s
19 genpdflem.s . . . . . . . . . 10  s  s  Q.
2019ex 108 . . . . . . . . 9  s  s  Q.
2120pm4.71rd 374 . . . . . . . 8  s  s  Q.  s
2221anbi1d 438 . . . . . . 7  s  q  r G s  s  Q.  s  q 
r G s
2322exbidv 1703 . . . . . 6  s s  q  r G s  s s  Q.  s  q 
r G s
24 df-rex 2306 . . . . . . 7  s  Q. 
s  q  r G s  s s  Q. 
s  q  r G s
25 anass 381 . . . . . . . 8  s  Q.  s  q 
r G s  s 
Q.  s  q  r G s
2625exbii 1493 . . . . . . 7  s s  Q.  s  q  r G s  s s  Q.  s  q 
r G s
2724, 26bitr4i 176 . . . . . 6  s  Q. 
s  q  r G s  s s  Q.  s  q 
r G s
2823, 27syl6rbbr 188 . . . . 5  s 
Q.  s  q  r G s  s s  q 
r G s
29 df-rex 2306 . . . . 5  s  q  r G s  s s  q  r G s
3028, 29syl6bbr 187 . . . 4  s 
Q.  s  q  r G s  s  q  r G s
3130rexbidv 2321 . . 3  r  s 
Q.  s  q  r G s  r  s  q  r G s
3218, 31bitrd 177 . 2  r 
Q.  s  Q.  r  s  q 
r G s  r  s  q  r G s
3332rabbidv 2543 1  { q  Q.  |  r  Q.  s  Q. 
r  s  q  r G s }  { q  Q.  |  r  s  q  r G s }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 884   wceq 1242  wex 1378   wcel 1390  wrex 2301   {crab 2304  (class class class)co 5455   Q.cnq 6264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-ral 2305  df-rex 2306  df-rab 2309
This theorem is referenced by:  genpdf  6491
  Copyright terms: Public domain W3C validator