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Theorem ssrabeq 3122
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 3121 . . 3  |-  { x  e.  V  |  ph }  C_  V
21biantru 297 . 2  |-  ( V 
C_  { x  e.  V  |  ph }  <->  ( V  C_  { x  e.  V  |  ph }  /\  { x  e.  V  |  ph }  C_  V
) )
3 eqss 3054 . 2  |-  ( V  =  { x  e.  V  |  ph }  <->  ( V  C_  { x  e.  V  |  ph }  /\  { x  e.  V  |  ph }  C_  V
) )
42, 3bitr4i 186 1  |-  ( V 
C_  { x  e.  V  |  ph }  <->  V  =  { x  e.  V  |  ph }
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1296   {crab 2374    C_ wss 3013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rab 2379  df-in 3019  df-ss 3026
This theorem is referenced by:  difrab0eqim  3368
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