Theorem sseldi 3272

Description: Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)

Hypotheses

Ref	Expression
sseli.1	⊢ A ⊆ B
sseldi.2	⊢ (φ → C ∈ A)

Assertion

Ref	Expression
sseldi	⊢ (φ → C ∈ B)

Proof of Theorem sseldi

Step	Hyp	Ref	Expression
1		sseldi.2	. 2 ⊢ (φ → C ∈ A)
2		sseli.1	. . 3 ⊢ A ⊆ B
3	2	sseli 3270	. 2 ⊢ (C ∈ A → C ∈ B)
4	1, 3	syl 15	1 ⊢ (φ → 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:  pw1eqadj  4333  txpcofun  5804