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Theorem dfss2f 2999
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 2997 . 2  |-  ( A 
C_  B  <->  A. z
( z  e.  A  ->  z  e.  B ) )
2 dfss2f.1 . . . . 5  |-  F/_ x A
32nfcri 2217 . . . 4  |-  F/ x  z  e.  A
4 dfss2f.2 . . . . 5  |-  F/_ x B
54nfcri 2217 . . . 4  |-  F/ x  z  e.  B
63, 5nfim 1505 . . 3  |-  F/ x
( z  e.  A  ->  z  e.  B )
7 nfv 1462 . . 3  |-  F/ z ( x  e.  A  ->  x  e.  B )
8 eleq1 2145 . . . 4  |-  ( z  =  x  ->  (
z  e.  A  <->  x  e.  A ) )
9 eleq1 2145 . . . 4  |-  ( z  =  x  ->  (
z  e.  B  <->  x  e.  B ) )
108, 9imbi12d 232 . . 3  |-  ( z  =  x  ->  (
( z  e.  A  ->  z  e.  B )  <-> 
( x  e.  A  ->  x  e.  B ) ) )
116, 7, 10cbval 1679 . 2  |-  ( A. z ( z  e.  A  ->  z  e.  B )  <->  A. x
( x  e.  A  ->  x  e.  B ) )
121, 11bitri 182 1  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    e. wcel 1434   F/_wnfc 2210    C_ wss 2982
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 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-in 2988  df-ss 2995
This theorem is referenced by:  dfss3f  3000  ssrd  3013  ss2ab  3071
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