Theorem ssrind 3408

Description: Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
ssrind.1	|- ( ph -> A C_ B )

Assertion

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

Proof of Theorem ssrind

Step	Hyp	Ref	Expression
1		ssrind.1	. 2 |- ( ph -> A C_ B )
2		ssrin 3406	. 2 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A i^i C ) C_ ( B i^i C ) )

Syntax hints:   -> wi 4   i^i cin 3173   C_ wss 3174

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187

This theorem is referenced by:  restbasg  14755