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Theorem eloprabi 6874
 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 2475 . . . . . 6
21anbi1d 719 . . . . 5
323exbidv 1779 . . . 4
4 df-oprab 6312 . . . 4
53, 4elab2g 3175 . . 3
65ibi 249 . 2
7 opex 4664 . . . . . . . . . . 11
8 vex 3034 . . . . . . . . . . 11
97, 8op1std 6822 . . . . . . . . . 10
109fveq2d 5883 . . . . . . . . 9
11 vex 3034 . . . . . . . . . 10
12 vex 3034 . . . . . . . . . 10
1311, 12op1st 6820 . . . . . . . . 9
1410, 13syl6req 2522 . . . . . . . 8
15 eloprabi.1 . . . . . . . 8
1614, 15syl 17 . . . . . . 7
179fveq2d 5883 . . . . . . . . 9
1811, 12op2nd 6821 . . . . . . . . 9
1917, 18syl6req 2522 . . . . . . . 8
20 eloprabi.2 . . . . . . . 8
2119, 20syl 17 . . . . . . 7
227, 8op2ndd 6823 . . . . . . . . 9
2322eqcomd 2477 . . . . . . . 8
24 eloprabi.3 . . . . . . . 8
2523, 24syl 17 . . . . . . 7
2616, 21, 253bitrd 287 . . . . . 6
2726biimpa 492 . . . . 5
2827exlimiv 1784 . . . 4
2928exlimiv 1784 . . 3
3029exlimiv 1784 . 2
316, 30syl 17 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   wceq 1452  wex 1671   wcel 1904  cop 3965  cfv 5589  coprab 6309  c1st 6810  c2nd 6811 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-oprab 6312  df-1st 6812  df-2nd 6813 This theorem is referenced by: (None)
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