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Theorem eloprabi 6415
 Description: A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
eloprabi.1
eloprabi.2
eloprabi.3
Assertion
Ref Expression
eloprabi
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)   (,,)

Proof of Theorem eloprabi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2444 . . . . . 6
21anbi1d 687 . . . . 5
323exbidv 1640 . . . 4
4 df-oprab 6087 . . . 4
53, 4elab2g 3086 . . 3
65ibi 234 . 2
7 opex 4429 . . . . . . . . . . 11
8 vex 2961 . . . . . . . . . . 11
97, 8op1std 6359 . . . . . . . . . 10
109fveq2d 5734 . . . . . . . . 9
11 vex 2961 . . . . . . . . . 10
12 vex 2961 . . . . . . . . . 10
1311, 12op1st 6357 . . . . . . . . 9
1410, 13syl6req 2487 . . . . . . . 8
15 eloprabi.1 . . . . . . . 8
1614, 15syl 16 . . . . . . 7
179fveq2d 5734 . . . . . . . . 9
1811, 12op2nd 6358 . . . . . . . . 9
1917, 18syl6req 2487 . . . . . . . 8
20 eloprabi.2 . . . . . . . 8
2119, 20syl 16 . . . . . . 7
227, 8op2ndd 6360 . . . . . . . . 9
2322eqcomd 2443 . . . . . . . 8
24 eloprabi.3 . . . . . . . 8
2523, 24syl 16 . . . . . . 7
2616, 21, 253bitrd 272 . . . . . 6
2726biimpa 472 . . . . 5
2827exlimiv 1645 . . . 4
2928exlimiv 1645 . . 3
3029exlimiv 1645 . 2
316, 30syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wex 1551   wceq 1653   wcel 1726  cop 3819  cfv 5456  coprab 6084  c1st 6349  c2nd 6350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fv 5464  df-oprab 6087  df-1st 6351  df-2nd 6352
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