Theorem sseqtrrid 3148

Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
sseqtrrid.1	|- B C_ A
sseqtrrid.2	|- ( ph -> C = A )

Assertion

Ref	Expression
sseqtrrid	|- ( ph -> B C_ C )

Proof of Theorem sseqtrrid

Step	Hyp	Ref	Expression
1		sseqtrrid.1	. . 3 |- B C_ A
2	1	a1i 9	. 2 |- ( ph -> B C_ A )
3		sseqtrrid.2	. 2 |- ( ph -> C = A )
4	2, 3	sseqtrrd 3136	1 |- ( ph -> B C_ C )

Syntax hints:   -> wi 4   = wceq 1331   C_ wss 3071

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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084

This theorem is referenced by:  resdif  5389  fimacnv  5549  tfrlem5  6211  fsumsplit  11176  phimullem  11901  ennnfonelemss  11923  istopon  12180  sscls  12289  mopnfss  12616