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Theorem dfss3f 3342
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 3341 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
4 df-ral 2712 . 2  |-  ( A. x  e.  A  x  e.  B  <->  A. x ( x  e.  A  ->  x  e.  B ) )
53, 4bitr4i 245 1  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550    e. wcel 1726   F/_wnfc 2561   A.wral 2707    C_ wss 3322
This theorem is referenced by:  nfss  3343  sigaclcu2  24508  heibor1  26533  stoweidlem52  27791  bnj1498  29504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-in 3329  df-ss 3336
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