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Theorem dfss2f 3058
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 3056 . 2  |-  ( A 
C_  B  <->  A. z
( z  e.  A  ->  z  e.  B ) )
2 dfss2f.1 . . . . 5  |-  F/_ x A
32nfcri 2252 . . . 4  |-  F/ x  z  e.  A
4 dfss2f.2 . . . . 5  |-  F/_ x B
54nfcri 2252 . . . 4  |-  F/ x  z  e.  B
63, 5nfim 1536 . . 3  |-  F/ x
( z  e.  A  ->  z  e.  B )
7 nfv 1493 . . 3  |-  F/ z ( x  e.  A  ->  x  e.  B )
8 eleq1 2180 . . . 4  |-  ( z  =  x  ->  (
z  e.  A  <->  x  e.  A ) )
9 eleq1 2180 . . . 4  |-  ( z  =  x  ->  (
z  e.  B  <->  x  e.  B ) )
108, 9imbi12d 233 . . 3  |-  ( z  =  x  ->  (
( z  e.  A  ->  z  e.  B )  <-> 
( x  e.  A  ->  x  e.  B ) ) )
116, 7, 10cbval 1712 . 2  |-  ( A. z ( z  e.  A  ->  z  e.  B )  <->  A. x
( x  e.  A  ->  x  e.  B ) )
121, 11bitri 183 1  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1314    e. wcel 1465   F/_wnfc 2245    C_ wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-in 3047  df-ss 3054
This theorem is referenced by:  dfss3f  3059  ssrd  3072  ssrmof  3130  ss2ab  3135
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