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Theorem rabss2 4087
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 1807 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) → ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
3 df-ss 3979 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
4 ss2ab 4071 . . 3 ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
52, 3, 43imtr4i 292 . 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 4040 1 (𝐴𝐵 → {𝑥𝐴𝜑} ⊆ {𝑥𝐵𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1534  wcel 2105  {cab 2711  {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-rab 3433  df-ss 3979
This theorem is referenced by:  rabssrabd  4092  sess2  5654  hashbcss  17037  dprdss  20063  minveclem4  25479  prmdvdsfi  27164  mumul  27238  sqff1o  27239  rpvmasumlem  27545  disjxwwlkn  29942  clwwlknfi  30073  shatomistici  32389  rabfodom  32532  xpinpreima2  33867  ballotth  34518  bj-unrab  36908  icorempo  37333  lssats  38993  lpssat  38994  lssatle  38996  lssat  38997  atlatmstc  39300  dochspss  41360  unitscyglem4  42179  rmxyelqirrOLD  42898  idomodle  43179  sssmf  46693
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