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Theorem ss2rabdv 4072
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 3144 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
3 ss2rab 4067 . 2 ({𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒} ↔ ∀𝑥𝐴 (𝜓𝜒))
42, 3sylibr 233 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2104  wral 3059  {crab 3430  wss 3947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rab 3431  df-v 3474  df-in 3954  df-ss 3964
This theorem is referenced by:  rabssrabd  4080  sess1  5643  suppssov1  8185  suppssfv  8189  cofon1  8673  naddssim  8686  harword  9560  scottex  9882  mrcss  17564  ablfac1b  19981  mptscmfsupp0  20681  lspss  20739  dsmmacl  21515  dsmmsubg  21517  dsmmlss  21518  aspss  21650  scmatdmat  22237  clsss  22778  lfinpfin  23248  qustgpopn  23844  metss2lem  24240  equivcau  25048  rrxmvallem  25152  ovolsslem  25233  itg2monolem1  25500  lgamucov  26778  sqff1o  26922  musum  26931  madess  27608  cofcut1  27645  cusgrfilem1  28979  clwlknf1oclwwlknlem3  29603  occon  30807  spanss  30868  rmfsupp2  32657  fldgenss  32676  locfinreflem  33118  omsmon  33595  orvclteinc  33772  fin2solem  36777  poimirlem26  36817  poimirlem27  36818  cnambfre  36839  pclssN  39068  2polssN  39089  dihglblem3N  40469  dochss  40539  mapdordlem2  40811  nna4b4nsq  41704  itgoss  42207  nzss  43378  ovnsslelem  45574  rmsuppss  47134  mndpsuppss  47135  scmsuppss  47136
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