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Theorem rabeq 2729
Description: Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
rabeq  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem rabeq
StepHypRef Expression
1 nfcv 2319 . 2  |-  F/_ x A
2 nfcv 2319 . 2  |-  F/_ x B
31, 2rabeqf 2727 1  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   {crab 2459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464
This theorem is referenced by:  rabeqdv  2731  rabeqbidv  2732  rabeqbidva  2733  difeq1  3246  ifeq1  3537  ifeq2  3538  elfvmptrab  5606  pmvalg  6652  unfiexmid  6910  ssfirab  6926  supeq2  6981  iooval2  9889  fzval2  9985  lcmval  12033  lcmcllem  12037  lcmledvds  12040  clsfval  13234
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