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Theorem elabrex 5429
 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:   ,   ,,
Allowed substitution hint:   ()

Proof of Theorem elabrex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 tru 1289 . . . 4
2 csbeq1a 2917 . . . . . . 7
32equcoms 1635 . . . . . 6
4 a1tru 1301 . . . . . 6
53, 42thd 173 . . . . 5
65rspcev 2702 . . . 4
71, 6mpan2 416 . . 3
8 elabrex.1 . . . 4
9 eqeq1 2088 . . . . 5
109rexbidv 2370 . . . 4
118, 10elab 2739 . . 3
127, 11sylibr 132 . 2
13 nfv 1462 . . . 4
14 nfcsb1v 2939 . . . . 5
1514nfeq2 2231 . . . 4
162eqeq2d 2093 . . . 4
1713, 15, 16cbvrex 2575 . . 3
1817abbii 2195 . 2
1912, 18syl6eleqr 2173 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1285   wtru 1286   wcel 1434  cab 2068  wrex 2350  cvv 2602  csb 2909 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910 This theorem is referenced by:  eusvobj2  5529
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