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Theorem isplig 21765
 Description: The predicate "is a planar incidence geometry". (Contributed by FL, 2-Aug-2009.)
Hypothesis
Ref Expression
isplig.1
Assertion
Ref Expression
isplig
Distinct variable groups:   ,,,,   ,,,
Allowed substitution hints:   (,,,)   ()

Proof of Theorem isplig
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 unieq 4024 . . . . 5
2 isplig.1 . . . . 5
31, 2syl6eqr 2486 . . . 4
4 reueq1 2906 . . . . . 6
54imbi2d 308 . . . . 5
63, 5raleqbidv 2916 . . . 4
73, 6raleqbidv 2916 . . 3
83rexeqdv 2911 . . . . 5
93, 8rexeqbidv 2917 . . . 4
109raleqbi1dv 2912 . . 3
11 raleq 2904 . . . . . 6
123, 11rexeqbidv 2917 . . . . 5
133, 12rexeqbidv 2917 . . . 4
143, 13rexeqbidv 2917 . . 3
157, 10, 143anbi123d 1254 . 2
16 df-plig 21764 . 2
1715, 16elab2g 3084 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725   wne 2599  wral 2705  wrex 2706  wreu 2707  cuni 4015  cplig 21763 This theorem is referenced by:  tncp  21766 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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-v 2958  df-uni 4016  df-plig 21764
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