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Theorem rabbidv 2566
Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
Hypothesis
Ref Expression
rabbidv.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
rabbidv  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  A  |  ch } )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)

Proof of Theorem rabbidv
StepHypRef Expression
1 rabbidv.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21adantr 265 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ch ) )
32rabbidva 2565 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  A  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259    e. wcel 1409   {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-ral 2328  df-rab 2332
This theorem is referenced by:  rabeqbidv  2569  difeq2  3084  seex  4100  mptiniseg  4843  supeq1  6392  supeq2  6395  supeq3  6396  cardcl  6419  isnumi  6420  cardval3ex  6423  carden2bex  6427  genpdflem  6663  genipv  6665  genpelxp  6667  addcomprg  6734  mulcomprg  6736  uzval  8571  ixxval  8866  fzval  8978  shftfn  9653
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