Theorem rabss2 4030

Description: Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid axioms. (Revised by TM, 1-Feb-2026.)

Assertion

Ref	Expression
rabss2	⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabss2

Step	Hyp	Ref	Expression
1		ssel 3930	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	anim1d 620	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
3	2	ss2abdv 4018	. 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
4		df-rab 3415	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
5		df-rab 3415	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
6	3, 4, 5	3sstr4g 3989	1 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142  {cab 2740  {crab 3414   ⊆ wss 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-ss 3921

This theorem is referenced by:  rabssrabd  4036  sess2  5613  hashbcss  17040  dprdss  20071  minveclem4  25491  prmdvdsfi  27168  mumul  27242  sqff1o  27243  rpvmasumlem  27548  disjxwwlkn  30110  clwwlknfi  30244  shatomistici  32561  rabfodom  32701  xpinpreima2  34201  ballotth  34832  bj-unrab  37408  icorempo  37842  lssats  39633  lpssat  39634  lssatle  39636  lssat  39637  atlatmstc  39940  dochspss  41999  unitscyglem4  42812  idomodle  43765  sssmf  47309