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Theorem ss2rabdv 4085
Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
Hypothesis
Ref Expression
ss2rabdv.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ss2rabdv (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem ss2rabdv
StepHypRef Expression
1 ss2rabdv.1 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
21ralrimiva 3143 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
3 ss2rab 4080 . 2 ({𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒} ↔ ∀𝑥𝐴 (𝜓𝜒))
42, 3sylibr 234 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105  wral 3058  {crab 3432  wss 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rab 3433  df-ss 3979
This theorem is referenced by:  rabssrabd  4092  sess1  5653  suppssov1  8220  suppssov2  8221  suppssfv  8225  cofon1  8708  naddssim  8721  harword  9600  scottex  9922  mrcss  17660  mndpsuppss  18790  ablfac1b  20104  mptscmfsupp0  20941  lspss  20999  dsmmacl  21778  dsmmsubg  21780  dsmmlss  21781  aspss  21914  psdmul  22187  scmatdmat  22536  clsss  23077  lfinpfin  23547  qustgpopn  24143  metss2lem  24539  equivcau  25347  rrxmvallem  25451  ovolsslem  25532  itg2monolem1  25799  lgamucov  27095  sqff1o  27239  musum  27248  madess  27929  cofcut1  27968  cusgrfilem1  29487  clwlknf1oclwwlknlem3  30111  occon  31315  spanss  31376  rmfsupp2  33227  fldgenss  33297  locfinreflem  33800  omsmon  34279  orvclteinc  34456  fin2solem  37592  poimirlem26  37632  poimirlem27  37633  cnambfre  37654  pclssN  39876  2polssN  39897  dihglblem3N  41277  dochss  41347  mapdordlem2  41619  nna4b4nsq  42646  itgoss  43151  nzss  44312  ovnsslelem  46515  gpgusgralem  47945  rmsuppss  48214  scmsuppss  48215
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