Theorem eqsstrdi 3207

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 3191	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1353   C_ wss 3129

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142

This theorem is referenced by:  eqsstrrdi  3208  resasplitss  5392  fimacnv  5642  en2other2  7190  exmidfodomrlemim  7195  pw1on  7220  suplocexprlemex  7716  1arith  12355  ennnfonelemkh  12403  aprap  13243  toponsspwpwg  13302  ntrss2  13403  cnprcl2k  13488  reldvg  13930  bj-nntrans  14474  nninfsellemsuc  14532