Theorem syl6sseq 2991

Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
syl6sseq.1	|- ( ph -> A C_ B )
syl6sseq.2	|- B = C

Assertion

Ref	Expression
syl6sseq	|- ( ph -> A C_ C )

Proof of Theorem syl6sseq

Step	Hyp	Ref	Expression
1		syl6sseq.1	. 2 |- ( ph -> A C_ B )
2		syl6sseq.2	. . 3 |- B = C
3	2	sseq2i 2970	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	sylib 127	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1243   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by:  syl6sseqr  2992  onintonm  4243  relrelss  4844  iotanul  4882  foimacnv  5144  cauappcvgprlemladdru  6754