Theorem ssind 3480

Description: A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
ssind.1	⊢ (φ → A ⊆ B)
ssind.2	⊢ (φ → A ⊆ C)

Assertion

Ref	Expression
ssind	⊢ (φ → A ⊆ (B ∩ C))

Proof of Theorem ssind

Step	Hyp	Ref	Expression
1		ssind.1	. 2 ⊢ (φ → A ⊆ B)
2		ssind.2	. 2 ⊢ (φ → A ⊆ C)
3		ssin 3478	. . 3 ⊢ ((A ⊆ B ∧ A ⊆ C) ↔ A ⊆ (B ∩ C))
4	3	biimpi 186	. 2 ⊢ ((A ⊆ B ∧ A ⊆ C) → A ⊆ (B ∩ C))
5	1, 2, 4	syl2anc 642	1 ⊢ (φ → A ⊆ (B ∩ C))

Syntax hints:   → wi 4   ∧ wa 358   ∩ cin 3209   ⊆ 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: (None)