Theorem eqsstrdi 3300

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 3284	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1402   C_ wss 3220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  eqsstrrdi  3301  resasplitss  5564  fimacnv  5828  suppssdmg  6479  en2other2  7538  exmidfodomrlemim  7543  pw1on  7575  suplocexprlemex  8079  fzowrddc  11397  swrdlend  11408  1arith  13124  ennnfonelemkh  13281  aprap  14571  znf1o  14958  mplbasss  15010  toponsspwpwg  15046  ntrss2  15145  cnprcl2k  15230  reldvg  15703  uhgrspansubgr  16432  trlsex  16542  bj-nntrans  16891  nninfsellemsuc  16960