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Theorem elabrex 6155
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
Hypothesis
Ref Expression
elabrex.1
Assertion
Ref Expression
elabrex
Distinct variable groups:   ,   , ,

Proof of Theorem elabrex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 tru 1399 . . . 4
2 csbeq1a 3443 . . . . . . 7
32equcoms 1795 . . . . . 6
4 a1tru 1411 . . . . . 6
53, 42thd 240 . . . . 5
65rspcev 3210 . . . 4
71, 6mpan2 671 . . 3
8 elabrex.1 . . . 4
9 eqeq1 2461 . . . . 5
109rexbidv 2968 . . . 4
118, 10elab 3246 . . 3
127, 11sylibr 212 . 2
13 nfv 1707 . . . 4
14 nfcsb1v 3450 . . . . 5
1514nfeq2 2636 . . . 4
162eqeq2d 2471 . . . 4
1713, 15, 16cbvrex 3081 . . 3
1817abbii 2591 . 2
1912, 18syl6eleqr 2556 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  =wceq 1395   wtru 1396  e.wcel 1818  {cab 2442  E.wrex 2808   cvv 3109  [_csb 3434
This theorem is referenced by:  eusvobj2  6289  lss1d  17609  prdsxmetlem  20871  prdsbl  20994  itg2monolem1  22157  heibor1  30306  dihglblem5  37025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-csb 3435
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