Theorem syl6sseq 3072

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 3051	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	sylib 120	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1289   C_ wss 2999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012

This theorem is referenced by:  syl6sseqr  3073  onintonm  4334  relrelss  4957  iotanul  4995  foimacnv  5271  cauappcvgprlemladdru  7215  zisum  10774  fsum3cvg3  10789