Theorem ssabral 3996

Description: The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)

Assertion

Ref	Expression
ssabral	⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssabral

Step	Hyp	Ref	Expression
1		ssab 3995	. 2 ⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		df-ral 3069	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3	1, 2	bitr4i 277	1 ⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   ∈ wcel 2106  {cab 2715  ∀wral 3064   ⊆ wss 3887

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-v 3434  df-in 3894  df-ss 3904

This theorem is referenced by:  comppfsc  22683  txdis1cn  22786  qustgplem  23272  xrhmeo  24109  cncmet  24486  itg1addlem4  24863  itg1addlem4OLD  24864  subfacp1lem6  33147  poimirlem9  35786  istotbnd3  35929  sstotbnd  35933  heibor1lem  35967  heibor1  35968  setis  46403