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Theorem elabrex 5986
 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 1331 . . . 4
2 csbeq1a 3260 . . . . . . 7
32equcoms 1694 . . . . . 6
4 a1tru 1340 . . . . . 6
53, 42thd 233 . . . . 5
65rspcev 3053 . . . 4
71, 6mpan2 654 . . 3
8 elabrex.1 . . . 4
9 eqeq1 2443 . . . . 5
109rexbidv 2727 . . . 4
118, 10elab 3083 . . 3
127, 11sylibr 205 . 2
13 nfv 1630 . . . 4
14 nfcsb1v 3284 . . . . 5
1514nfeq2 2584 . . . 4
162eqeq2d 2448 . . . 4
1713, 15, 16cbvrex 2930 . . 3
1817abbii 2549 . 2
1912, 18syl6eleqr 2528 1
 Colors of variables: wff set class Syntax hints:   wi 4   wtru 1326   wceq 1653   wcel 1726  cab 2423  wrex 2707  cvv 2957  csb 3252 This theorem is referenced by:  eusvobj2  6583  lss1d  16040  prdsxmetlem  18399  prdsbl  18522  itg2monolem1  19643  heibor1  26520  dihglblem5  32097 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rex 2712  df-v 2959  df-sbc 3163  df-csb 3253
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