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Theorem ssext 4244
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 4243 . . 3  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
2 ssextss 4243 . . 3  |-  ( B 
C_  A  <->  A. x
( x  C_  B  ->  x  C_  A )
)
31, 2anbi12i 678 . 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 3207 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
5 albiim 1601 . 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 268 1  |-  ( A  =  B  <->  A. x
( x  C_  A  <->  x 
C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    C_ wss 3165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-pr 3660
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