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Theorem ss2rabi 3663
 Description: Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
Hypothesis
Ref Expression
ss2rabi.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ss2rabi {𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓}

Proof of Theorem ss2rabi
StepHypRef Expression
1 ss2rab 3657 . 2 ({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))
2 ss2rabi.1 . 2 (𝑥𝐴 → (𝜑𝜓))
31, 2mprgbir 2922 1 {𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1987  {crab 2911   ⊆ wss 3555 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-in 3562  df-ss 3569 This theorem is referenced by:  supub  8309  suplub  8310  card2on  8403  rankval4  8674  fin1a2lem12  9177  catlid  16265  catrid  16266  gsumval2  17201  lbsextlem3  19079  psrbagsn  19414  musum  24817  ppiub  24829  umgrupgr  25893  umgrislfupgr  25913  usgruspgr  25966  usgrislfuspgr  25972  disjxwwlksn  26668  clwwlknclwwlkdifnum  26741  konigsbergssiedgw  26977  omssubadd  30143  bj-unrab  32569  poimirlem26  33067  poimirlem27  33068  lclkrs2  36309
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