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Theorem abexex 5782
 Description: A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
Hypotheses
Ref Expression
abexex.1
abexex.2
abexex.3
Assertion
Ref Expression
abexex
Distinct variable group:   ,,
Allowed substitution hints:   (,)

Proof of Theorem abexex
StepHypRef Expression
1 df-rex 2549 . . . 4
2 abexex.2 . . . . . 6
32pm4.71ri 614 . . . . 5
43exbii 1569 . . . 4
51, 4bitr4i 243 . . 3
65abbii 2395 . 2
7 abexex.1 . . 3
8 abexex.3 . . 3
97, 8abrexex2 5780 . 2
106, 9eqeltrri 2354 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358  wex 1528   wcel 1684  cab 2269  wrex 2544  cvv 2788 This theorem is referenced by:  brdom7disj  8156  brdom6disj  8157 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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