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Theorem eloprabi 6094
 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 2146 . . . . . 6
21anbi1d 460 . . . . 5
323exbidv 1841 . . . 4
4 df-oprab 5778 . . . 4
53, 4elab2g 2831 . . 3
65ibi 175 . 2
7 vex 2689 . . . . . . . . . . . 12
8 vex 2689 . . . . . . . . . . . 12
97, 8opex 4151 . . . . . . . . . . 11
10 vex 2689 . . . . . . . . . . 11
119, 10op1std 6046 . . . . . . . . . 10
1211fveq2d 5425 . . . . . . . . 9
137, 8op1st 6044 . . . . . . . . 9
1412, 13syl6req 2189 . . . . . . . 8
15 eloprabi.1 . . . . . . . 8
1614, 15syl 14 . . . . . . 7
1711fveq2d 5425 . . . . . . . . 9
187, 8op2nd 6045 . . . . . . . . 9
1917, 18syl6req 2189 . . . . . . . 8
20 eloprabi.2 . . . . . . . 8
2119, 20syl 14 . . . . . . 7
229, 10op2ndd 6047 . . . . . . . . 9
2322eqcomd 2145 . . . . . . . 8
24 eloprabi.3 . . . . . . . 8
2523, 24syl 14 . . . . . . 7
2616, 21, 253bitrd 213 . . . . . 6
2726biimpa 294 . . . . 5
2827exlimiv 1577 . . . 4
2928exlimiv 1577 . . 3
3029exlimiv 1577 . 2
316, 30syl 14 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1331  wex 1468   wcel 1480  cop 3530  cfv 5123  coprab 5775  c1st 6036  c2nd 6037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-oprab 5778  df-1st 6038  df-2nd 6039 This theorem is referenced by: (None)
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