Theorem eqsstrdi 3208

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 3192	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1353   C_ wss 3130

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 3136  df-ss 3143

This theorem is referenced by:  eqsstrrdi  3209  resasplitss  5396  fimacnv  5646  en2other2  7195  exmidfodomrlemim  7200  pw1on  7225  suplocexprlemex  7721  1arith  12365  ennnfonelemkh  12413  aprap  13376  toponsspwpwg  13525  ntrss2  13624  cnprcl2k  13709  reldvg  14151  bj-nntrans  14706  nninfsellemsuc  14764