Theorem 3sstr4d 3293

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
3sstr4d.1	|- ( ph -> A C_ B )
3sstr4d.2	|- ( ph -> C = A )
3sstr4d.3	|- ( ph -> D = B )

Assertion

Ref	Expression
3sstr4d	|- ( ph -> C C_ D )

Proof of Theorem 3sstr4d

Step	Hyp	Ref	Expression
1		3sstr4d.1	. 2 |- ( ph -> A C_ B )
2		3sstr4d.2	. . 3 |- ( ph -> C = A )
3		3sstr4d.3	. . 3 |- ( ph -> D = B )
4	2, 3	sseq12d 3279	. 2 |- ( ph -> ( C C_ D <-> A C_ B ) )
5	1, 4	mpbird 167	1 |- ( ph -> C C_ D )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   C_ wss 3220

This proof depends on 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 proof 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 used by:  ressuppss  6494  suppfnss  6497  suppssfvg  6503  rdgss  6654  sucinc2  6719  oawordi  6742  nnnninf  7466  fzoss1  10590  fzoss2  10591  swrd0g  11446  lspss  14785  aspss  15068  clsss  15268  ntrss  15269  sslm  15397  txss12  15416  metss2lem  15647  xmettxlem  15659  xmettx  15660  plyss  15888  ifpsnprss  16682  nnsf  17146  nninfself  17154