Theorem eqsstrdi 3232

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 3216	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3154

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3160  df-ss 3167

This theorem is referenced by:  eqsstrrdi  3233  resasplitss  5434  fimacnv  5688  en2other2  7258  exmidfodomrlemim  7263  pw1on  7288  suplocexprlemex  7784  1arith  12508  ennnfonelemkh  12572  aprap  13785  znf1o  14150  toponsspwpwg  14201  ntrss2  14300  cnprcl2k  14385  reldvg  14858  bj-nntrans  15513  nninfsellemsuc  15572