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Theorem ssrabeq 3500
Description: If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
Assertion
Ref Expression
ssrabeq  |-  ( V 
C_  { x  e.  V  |  ph }  <->  V  =  { x  e.  V  |  ph }
)
Distinct variable group:    x, V
Allowed substitution hint:    ph( x)

Proof of Theorem ssrabeq
StepHypRef Expression
1 ssrab2 3499 . . 3  |-  { x  e.  V  |  ph }  C_  V
21biantru 503 . 2  |-  ( V 
C_  { x  e.  V  |  ph }  <->  ( V  C_  { x  e.  V  |  ph }  /\  { x  e.  V  |  ph }  C_  V
) )
3 eqss 3432 . 2  |-  ( V  =  { x  e.  V  |  ph }  <->  ( V  C_  { x  e.  V  |  ph }  /\  { x  e.  V  |  ph }  C_  V
) )
42, 3bitr4i 252 1  |-  ( V 
C_  { x  e.  V  |  ph }  <->  V  =  { x  e.  V  |  ph }
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1399   {crab 2736    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-rab 2741  df-in 3396  df-ss 3403
This theorem is referenced by:  difrab0eq  3803
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