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Theorem sbcbidv 3045
Description: Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
Hypothesis
Ref Expression
sbcbidv.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
sbcbidv  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. A  /  x ]. ch ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)

Proof of Theorem sbcbidv
StepHypRef Expression
1 nfv 1605 . 2  |-  F/ x ph
2 sbcbidv.1 . 2  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2sbcbid 3044 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. A  /  x ]. ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   [.wsbc 2991
This theorem is referenced by:  sbcbii  3046  csbcomg  3104  opelopabsb  4275  opelopabf  4289  fpwwe2cbv  8252  fpwwe2lem2  8254  fpwwe2lem3  8255  isprs  14064  isdrs  14068  istos  14141  isdlat  14296  islmod  15631  elmptrab  17522  sbcbidv2  24969  bisig0  26062  isibg2  26110  isibcg  26191  indexa  26412  sdclem2  26452  sdclem1  26453  fdc  26455  sbccomieg  26870  rexrabdioph  26875  rexfrabdioph  26876  2rexfrabdioph  26877  3rexfrabdioph  26878  4rexfrabdioph  26879  6rexfrabdioph  26880  7rexfrabdioph  26881  2sbc6g  27615  2sbc5g  27616  hdmap1ffval  31986  hdmap1fval  31987  hdmapffval  32019  hdmapfval  32020  hgmapffval  32078  hgmapfval  32079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-sbc 2992
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