Theorem eqsstrdi 3176

Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

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

Assertion

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

Proof of Theorem eqsstrdi

Step	Hyp	Ref	Expression
1		eqsstrdi.1	. 2 |- ( ph -> A = B )
2		eqsstrdi.2	. . 3 |- B C_ C
3	2	a1i 9	. 2 |- ( ph -> B C_ C )
4	1, 3	eqsstrd 3160	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1332   C_ wss 3098

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-11 1483  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-in 3104  df-ss 3111

This theorem is referenced by:  eqsstrrdi  3177  resasplitss  5342  fimacnv  5589  en2other2  7110  exmidfodomrlemim  7115  pw1on  7140  suplocexprlemex  7621  ennnfonelemkh  12092  toponsspwpwg  12359  ntrss2  12460  cnprcl2k  12545  reldvg  12987  bj-nntrans  13464  nninfsellemsuc  13525