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Theorem dfss3f 3409
Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)
Hypotheses
Ref Expression
dfss2f.1  |-  F/_ x A
dfss2f.2  |-  F/_ x B
Assertion
Ref Expression
dfss3f  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )

Proof of Theorem dfss3f
StepHypRef Expression
1 dfss2f.1 . . 3  |-  F/_ x A
2 dfss2f.2 . . 3  |-  F/_ x B
31, 2dfss2f 3408 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
4 df-ral 2737 . 2  |-  ( A. x  e.  A  x  e.  B  <->  A. x ( x  e.  A  ->  x  e.  B ) )
53, 4bitr4i 252 1  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1397    e. wcel 1826   F/_wnfc 2530   A.wral 2732    C_ wss 3389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-in 3396  df-ss 3403
This theorem is referenced by:  nfss  3410  sigaclcu2  28269  heibor1  30472  bnj1498  34464
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