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Theorem dfss2f 3171
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 3169 . 2  |-  ( A 
C_  B  <->  A. z
( z  e.  A  ->  z  e.  B ) )
2 dfss2f.1 . . . . 5  |-  F/_ x A
32nfcri 2413 . . . 4  |-  F/ x  z  e.  A
4 dfss2f.2 . . . . 5  |-  F/_ x B
54nfcri 2413 . . . 4  |-  F/ x  z  e.  B
63, 5nfim 1769 . . 3  |-  F/ x
( z  e.  A  ->  z  e.  B )
7 nfv 1605 . . 3  |-  F/ z ( x  e.  A  ->  x  e.  B )
8 eleq1 2343 . . . 4  |-  ( z  =  x  ->  (
z  e.  A  <->  x  e.  A ) )
9 eleq1 2343 . . . 4  |-  ( z  =  x  ->  (
z  e.  B  <->  x  e.  B ) )
108, 9imbi12d 311 . . 3  |-  ( z  =  x  ->  (
( z  e.  A  ->  z  e.  B )  <-> 
( x  e.  A  ->  x  e.  B ) ) )
116, 7, 10cbval 1924 . 2  |-  ( A. z ( z  e.  A  ->  z  e.  B )  <->  A. x
( x  e.  A  ->  x  e.  B ) )
121, 11bitri 240 1  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   F/_wnfc 2406    C_ wss 3152
This theorem is referenced by:  dfss3f  3172  ss2ab  3241  rankval4  7539  ballotth  23096  ssrd  23125  ssrmo  23148  rabexgfGS  23171  sigaclcuni  23479  dvcosre  27741  itgsinexplem1  27748  stoweidlem52  27801
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-in 3159  df-ss 3166
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