Theorem abssi 3313

Description: Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Hypothesis

Ref	Expression
abssi.1	⊢ (𝜑 → 𝑥 ∈ 𝐴)

Assertion

Ref	Expression
abssi	⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abssi

Step	Hyp	Ref	Expression
1		abssi.1	. . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
2	1	ss2abi 3310	. 2 ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴}
3		abid2 2355	. 2 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
4	2, 3	sseqtri 3272	1 ⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴

Syntax hints:   → wi 4   ∈ wcel 2203  {cab 2218   ⊆ wss 3211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-in 3217  df-ss 3224

This theorem is referenced by:  ssab2  3322  abf  3552  intab  3978  opabss  4174  relopabi  4880  exse2  5136  mpoexw  6409  tfrlem8  6549  frecabex  6629  fiprc  7057  fival  7257  nqprxx  7861  ltnqex  7864  gtnqex  7865  recexprlemell  7937  recexprlemelu  7938  recexprlempr  7947  4sqlem1  13086  topnex  14951  2sqlem7  15994