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Theorem dfss2f 3339
Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
dfss2f.1  |-  F/_ x A
dfss2f.2  |-  F/_ x B
Assertion
Ref Expression
dfss2f  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )

Proof of Theorem dfss2f
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfss2 3337 . 2  |-  ( A 
C_  B  <->  A. z
( z  e.  A  ->  z  e.  B ) )
2 dfss2f.1 . . . . 5  |-  F/_ x A
32nfcri 2566 . . . 4  |-  F/ x  z  e.  A
4 dfss2f.2 . . . . 5  |-  F/_ x B
54nfcri 2566 . . . 4  |-  F/ x  z  e.  B
63, 5nfim 1832 . . 3  |-  F/ x
( z  e.  A  ->  z  e.  B )
7 nfv 1629 . . 3  |-  F/ z ( x  e.  A  ->  x  e.  B )
8 eleq1 2496 . . . 4  |-  ( z  =  x  ->  (
z  e.  A  <->  x  e.  A ) )
9 eleq1 2496 . . . 4  |-  ( z  =  x  ->  (
z  e.  B  <->  x  e.  B ) )
108, 9imbi12d 312 . . 3  |-  ( z  =  x  ->  (
( z  e.  A  ->  z  e.  B )  <-> 
( x  e.  A  ->  x  e.  B ) ) )
116, 7, 10cbval 1982 . 2  |-  ( A. z ( z  e.  A  ->  z  e.  B )  <->  A. x
( x  e.  A  ->  x  e.  B ) )
121, 11bitri 241 1  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    e. wcel 1725   F/_wnfc 2559    C_ wss 3320
This theorem is referenced by:  dfss3f  3340  ssrd  3353  ss2ab  3411  rankval4  7793  ssrmo  23981  rabexgfGS  23987  ballotth  24795  dvcosre  27717  itgsinexplem1  27724  stoweidlem59  27784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-in 3327  df-ss 3334
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