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Theorem ssext 4359
Description: An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
Assertion
Ref Expression
ssext  |-  ( A  =  B  <->  A. x
( x  C_  A  <->  x 
C_  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem ssext
StepHypRef Expression
1 ssextss 4358 . . 3  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
2 ssextss 4358 . . 3  |-  ( B 
C_  A  <->  A. x
( x  C_  B  ->  x  C_  A )
)
31, 2anbi12i 679 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  <->  ( A. x ( x  C_  A  ->  x  C_  B
)  /\  A. x
( x  C_  B  ->  x  C_  A )
) )
4 eqss 3306 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 albiim 1618 . 2  |-  ( A. x ( x  C_  A 
<->  x  C_  B )  <->  ( A. x ( x 
C_  A  ->  x  C_  B )  /\  A. x ( x  C_  B  ->  x  C_  A
) ) )
63, 4, 53bitr4i 269 1  |-  ( A  =  B  <->  A. x
( x  C_  A  <->  x 
C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    C_ wss 3263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-pw 3744  df-sn 3763  df-pr 3764
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