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Theorem rabss2 4072
Description: Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
rabss2 (𝐴𝐵 → {𝑥𝐴𝜑} ⊆ {𝑥𝐵𝜑})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabss2
StepHypRef Expression
1 pm3.45 622 . . . 4 ((𝑥𝐴𝑥𝐵) → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
21alimi 1813 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) → ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
3 dfss2 3965 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
4 ss2ab 4053 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
52, 3, 43imtr4i 291 . 2 (𝐴𝐵 → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥 ∣ (𝑥𝐵𝜑)})
6 df-rab 3433 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
7 df-rab 3433 . 2 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
85, 6, 73sstr4g 4024 1 (𝐴𝐵 → {𝑥𝐴𝜑} ⊆ {𝑥𝐵𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539  wcel 2106  {cab 2709  {crab 3432  wss 3945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-in 3952  df-ss 3962
This theorem is referenced by:  rabssrabd  4078  sess2  5639  hashbcss  16921  dprdss  19860  minveclem4  24880  prmdvdsfi  26540  mumul  26614  sqff1o  26615  rpvmasumlem  26919  disjxwwlkn  29096  clwwlknfi  29227  shatomistici  31541  rabfodom  31671  xpinpreima2  32782  ballotth  33431  bj-unrab  35674  icorempo  36100  lssats  37751  lpssat  37752  lssatle  37754  lssat  37755  atlatmstc  38058  dochspss  40118  rmxyelqirrOLD  41484  idomodle  41773  sssmf  45291
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