Theorem ssind 3357

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	|- ( ph -> A C_ B )
ssind.2	|- ( ph -> A C_ C )

Assertion

Ref	Expression
ssind	|- ( ph -> A C_ ( B i^i C ) )

Proof of Theorem ssind

Step	Hyp	Ref	Expression
1		ssind.1	. 2 |- ( ph -> A C_ B )
2		ssind.2	. 2 |- ( ph -> A C_ C )
3		ssin 3355	. . 3 |- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) )
4	3	biimpi 120	. 2 |- ( ( A C_ B /\ A C_ C ) -> A C_ ( B i^i C ) )
5	1, 2, 4	syl2anc 411	1 |- ( ph -> A C_ ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 104   i^i cin 3126   C_ wss 3127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-in 3133  df-ss 3140

This theorem is referenced by:  ntrin  13193  lmss  13315