Theorem sseld 3273

Description: Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)

Hypothesis

Ref	Expression
sseld.1	⊢ (φ → A ⊆ B)

Assertion

Ref	Expression
sseld	⊢ (φ → (C ∈ A → C ∈ B))

Proof of Theorem sseld

Step	Hyp	Ref	Expression
1		sseld.1	. 2 ⊢ (φ → A ⊆ B)
2		ssel 3268	. 2 ⊢ (A ⊆ B → (C ∈ A → C ∈ B))
3	1, 2	syl 15	1 ⊢ (φ → (C ∈ A → C ∈ B))

Syntax hints:   → wi 4   ∈ 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:  sselda  3274  sseldd  3275  ssneld  3276  elelpwi  3733  findsd  4411  sfinltfin  4536  ssbrd  4681  opelf  5236  fun11iun  5306  fvimacnv  5404  ffvelrn  5416  dff3  5421  dff4  5422  enprmaplem3  6079  spacind  6288  dmfrec  6317