ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cgsex4g Unicode version

Theorem cgsex4g 2585
Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
Hypotheses
Ref Expression
cgsex4g.1  C  D
cgsex4g.2
Assertion
Ref Expression
cgsex4g  R  S  C  R  D  S
Distinct variable groups:   ,,,,   ,,,,   , C,,,   , D,,,   ,,,,
Allowed substitution hints:   (,,,)   (,,,)    R(,,,)    S(,,,)

Proof of Theorem cgsex4g
StepHypRef Expression
1 cgsex4g.2 . . . . 5
21biimpa 280 . . . 4
32exlimivv 1773 . . 3
43exlimivv 1773 . 2
5 elisset 2562 . . . . . . . 8  R
6 elisset 2562 . . . . . . . 8  S
75, 6anim12i 321 . . . . . . 7  R  S
8 eeanv 1804 . . . . . . 7
97, 8sylibr 137 . . . . . 6  R  S
10 elisset 2562 . . . . . . . 8  C  R  C
11 elisset 2562 . . . . . . . 8  D  S  D
1210, 11anim12i 321 . . . . . . 7  C  R  D  S  C  D
13 eeanv 1804 . . . . . . 7  C  D  C  D
1412, 13sylibr 137 . . . . . 6  C  R  D  S  C  D
159, 14anim12i 321 . . . . 5  R  S  C  R  D  S  C  D
16 ee4anv 1806 . . . . 5  C  D  C  D
1715, 16sylibr 137 . . . 4  R  S  C  R  D  S  C  D
18 cgsex4g.1 . . . . . 6  C  D
19182eximi 1489 . . . . 5  C  D
20192eximi 1489 . . . 4  C  D
2117, 20syl 14 . . 3  R  S  C  R  D  S
221biimprcd 149 . . . . . 6
2322ancld 308 . . . . 5
24232eximdv 1759 . . . 4
25242eximdv 1759 . . 3
2621, 25syl5com 26 . 2  R  S  C  R  D  S
274, 26impbid2 131 1  R  S  C  R  D  S
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390
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-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  copsex4g  3975  brecop  6132
  Copyright terms: Public domain W3C validator