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  6498  suppfnss  6501  suppssfvg  6507  rdgss  6659  sucinc2  6724  oawordi  6747  nnnninf  7472  fzoss1  10596  fzoss2  10597  swrd0g  11453  lspss  14844  aspss  15127  clsss  15361  ntrss  15362  sslm  15490  txss12  15509  metss2lem  15740  xmettxlem  15752  xmettx  15753  plyss  15981  ifpsnprss  16806  nnsf  17271  nninfself  17279