Theorem ssini 2233

Description: An inference showing that the a subclass of two classes is a subclass of their intersection.

Hypotheses

Ref	Expression
ssini.1	|- A (_ B
ssini.2	|- A (_ C

Assertion

Ref	Expression
ssini	|- A (_ (B i^i C)

Proof of Theorem ssini

Step	Hyp	Ref	Expression
1		ssini.1	. . 3 |- A (_ B
2		ssini.2	. . 3 |- A (_ C
3	1, 2	pm3.2i 285	. 2 |- (A (_ B /\ A (_ C)
4		ssin 2232	. 2 |- ((A (_ B /\ A (_ C) <-> A (_ (B i^i C))
5	3, 4	mpbi 189	1 |- A (_ (B i^i C)

Syntax hints:   /\ wa 223   i^i cin 2046   (_ wss 2047

This theorem is referenced by:  inv1 2299  chm1 9379  chdmm1 9400  chm0 9413  ledi 9459  lejdi 9461  mdslj2 10247  mdslmd2 10257  sumdmdlem2 10346

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053

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