Theorem ssrdv 3279

Description: Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)

Hypothesis

Ref	Expression
ssrdv.1	⊢ (φ → (x ∈ A → x ∈ B))

Assertion

Ref	Expression
ssrdv	⊢ (φ → A ⊆ B)

Distinct variable groups:   x,A   x,B   φ,x

Proof of Theorem ssrdv

Step	Hyp	Ref	Expression
1		ssrdv.1	. . 3 ⊢ (φ → (x ∈ A → x ∈ B))
2	1	alrimiv 1631	. 2 ⊢ (φ → ∀x(x ∈ A → x ∈ B))
3		dfss2 3263	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
4	2, 3	sylibr 203	1 ⊢ (φ → A ⊆ B)

Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 1710   ⊆ 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:  sscon  3401  ssdif  3402  unss1  3433  ssrin  3481  eq0rdv  3586  uniss  3913  intss1  3942  intmin  3947  intssuni  3949  iunss1  3981  iinss1  3982  ss2iun  3985  ssiun  4009  ssiun2  4010  iinss  4018  iinss2  4019  sspwb  4119  pwadjoin  4120  phi11lem1  4596  phi011lem1  4599  ssrel  4845  dmss  4907  dmcosseq  4974  ssrnres  5060  fun11iun  5306  chfnrn  5400  ffnfv  5428  enadjlem1  6060  enmap2lem5  6068  enmap1lem5  6074  enprmaplem5  6081  enprmaplem6  6082