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Theorem psubspi 30544
 Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012.)
Hypotheses
Ref Expression
psubspset.l
psubspset.j
psubspset.a
psubspset.s
Assertion
Ref Expression
psubspi
Distinct variable groups:   ,,   ,,   ,,   ,   ,,
Allowed substitution hints:   (,)   (,)   (,)   (,)

Proof of Theorem psubspi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 psubspset.l . . . . . 6
2 psubspset.j . . . . . 6
3 psubspset.a . . . . . 6
4 psubspset.s . . . . . 6
51, 2, 3, 4ispsubsp2 30543 . . . . 5
65simplbda 608 . . . 4
76ex 424 . . 3
8 breq1 4215 . . . . . 6
982rexbidv 2748 . . . . 5
10 eleq1 2496 . . . . 5
119, 10imbi12d 312 . . . 4
1211rspccv 3049 . . 3
137, 12syl6 31 . 2
14133imp1 1166 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705  wrex 2706   wss 3320   class class class wbr 4212  cfv 5454  (class class class)co 6081  cple 13536  cjn 14401  catm 30061  cpsubsp 30293 This theorem is referenced by:  psubspi2N  30545  paddidm  30638 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-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 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-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  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-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-psubsp 30300
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