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Theorem fates 25058
 Description: Equivalence of and in the case of quantifiers restricted to a singleton. (Contributed by FL, 1-Jun-2011.)
Hypothesis
Ref Expression
fates.1
Assertion
Ref Expression
fates
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem fates
StepHypRef Expression
1 fates.1 . . . . 5
21elexi 2810 . . . 4
32alexeq 2910 . . 3
4 elsn 3668 . . . . 5
54imbi1i 315 . . . 4
65albii 1556 . . 3
74anbi1i 676 . . . 4
87exbii 1572 . . 3
93, 6, 83bitr4i 268 . 2
10 df-ral 2561 . 2
11 df-rex 2562 . 2
129, 10, 113bitr4i 268 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wal 1530  wex 1531   wceq 1632   wcel 1696  wral 2556  wrex 2557  csn 3653 This theorem is referenced by:  fatesg  25059 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561  df-rex 2562  df-v 2803  df-sn 3659
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