Theorem eqsstrri 3180

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 2174	. 2 |- A = B
3		eqsstr3.2	. 2 |- B C_ C
4	2, 3	eqsstri 3179	1 |- A C_ C

Syntax hints:   = wceq 1348   C_ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134

This theorem is referenced by:  inss2  3348  dmv  4825  resasplitss  5375  ofrfval  6067  ofvalg  6068  ofrval  6069  off  6071  ofres  6073  ofco  6077  dftpos4  6240  smores2  6271  caseinj  7063  djuinj  7080  bcm1k  10683  bcpasc  10689