Theorem rabss2 4001

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 3903	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	anim1d 623	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
3	2	ss2abdv 3989	. 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)})
4		df-rab 3413	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
5		df-rab 3413	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}
6	3, 4, 5	3sstr4g 3962	1 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144  {cab 2738  {crab 3412   ⊆ wss 3896

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-ss 3897

This theorem is used by:  rabssrabd  4007  sess2  5617  hashbcss  17191  dprdss  20254  minveclem4  25762  prmdvdsfi  27443  mumul  27517  sqff1o  27518  rpvmasumlem  27823  disjxwwlkn  30511  clwwlknfi  30645  shatomistici  32972  rabfodom  33152  xpinpreima2  34589  ballotth  35220  bj-unrab  37935  icorempo  38383  lssats  40178  lpssat  40179  lssatle  40181  lssat  40182  atlatmstc  40485  dochspss  42544  unitscyglem4  43357  idomodle  44302  sssmf  47856