Theorem ssini 2236

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 2235	. 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 2049   (_ wss 2050

This theorem is referenced by:  inv1 2303  chm1 9374  chdmm1 9395  chm0 9408  ledi 9454  lejdi 9456  mdslj2 10242  mdslmd2 10252  sumdmdlem2 10341

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-ss 2056

Copyright terms: Public domain