Theorem eqsstrdi 3219

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 3203	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1363   C_ wss 3141

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-in 3147  df-ss 3154

This theorem is referenced by:  eqsstrrdi  3220  resasplitss  5407  fimacnv  5658  en2other2  7208  exmidfodomrlemim  7213  pw1on  7238  suplocexprlemex  7734  1arith  12378  ennnfonelemkh  12426  aprap  13439  toponsspwpwg  13762  ntrss2  13861  cnprcl2k  13946  reldvg  14388  bj-nntrans  14943  nninfsellemsuc  15002