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Theorem rabss2 4076
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 pm3.45 623 . . . 4 ((𝑥𝐴𝑥𝐵) → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
21alimi 1814 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) → ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
3 dfss2 3969 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
4 ss2ab 4057 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
52, 3, 43imtr4i 292 . 2 (𝐴𝐵 → {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥 ∣ (𝑥𝐵𝜑)})
6 df-rab 3434 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
7 df-rab 3434 . 2 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
85, 6, 73sstr4g 4028 1 (𝐴𝐵 → {𝑥𝐴𝜑} ⊆ {𝑥𝐵𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wcel 2107  {cab 2710  {crab 3433  wss 3949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966
This theorem is referenced by:  rabssrabd  4082  sess2  5646  hashbcss  16937  dprdss  19899  minveclem4  24949  prmdvdsfi  26611  mumul  26685  sqff1o  26686  rpvmasumlem  26990  disjxwwlkn  29167  clwwlknfi  29298  shatomistici  31614  rabfodom  31743  xpinpreima2  32887  ballotth  33536  bj-unrab  35806  icorempo  36232  lssats  37882  lpssat  37883  lssatle  37885  lssat  37886  atlatmstc  38189  dochspss  40249  rmxyelqirrOLD  41649  idomodle  41938  sssmf  45454
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