Theorem abssi 3342

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 3339	. 2 ⊢ {x ∣ φ} ⊆ {x ∣ x ∈ A}
3		abid2 2471	. 2 ⊢ {x ∣ x ∈ A} = A
4	2, 3	sseqtri 3304	1 ⊢ {x ∣ φ} ⊆ A

Syntax hints:   → wi 4   ∈ wcel 1710  {cab 2339   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  ssab2  3351  abf  3585  intab  3957  opkabssvvk  4209  fvclss  5463  mapsspw  6023  spacssnc  6285