Theorem ssrind 3355

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 3353	. 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 3121   C_ wss 3122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-v 2733  df-in 3128  df-ss 3135

This theorem is referenced by:  restbasg  13079