Theorem eqsstrri 3226

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)

Hypotheses

Ref	Expression
eqsstr3.1	|- B = A
eqsstr3.2	|- B C_ C

Assertion

Ref	Expression
eqsstrri	|- A C_ C

Proof of Theorem eqsstrri

Step	Hyp	Ref	Expression
1		eqsstr3.1	. . 3 |- B = A
2	1	eqcomi 2209	. 2 |- A = B
3		eqsstr3.2	. 2 |- B C_ C
4	2, 3	eqsstri 3225	1 |- A C_ C

Syntax hints:   = wceq 1373   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179

This theorem is referenced by:  inss2  3394  dmv  4895  resasplitss  5457  ofrfval  6169  ofvalg  6170  ofrval  6171  off  6173  ofres  6175  ofco  6179  dftpos4  6351  smores2  6382  caseinj  7193  djuinj  7210  bcm1k  10907  bcpasc  10913  nninfctlemfo  12394