Theorem eqsstrrd 3194

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

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

Assertion

Ref	Expression
eqsstrrd	|- ( ph -> A C_ C )

Proof of Theorem eqsstrrd

Step	Hyp	Ref	Expression
1		eqsstrrd.1	. . 3 |- ( ph -> B = A )
2	1	eqcomd 2183	. 2 |- ( ph -> A = B )
3		eqsstrrd.2	. 2 |- ( ph -> B C_ C )
4	2, 3	eqsstrd 3193	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1353   C_ wss 3131

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144

This theorem is referenced by:  ssxpbm  5066  ssxp1  5067  ssxp2  5068  suppssof1  6103  tfrlemiubacc  6334  tfr1onlemubacc  6350  tfrcllemubacc  6363  oaword1  6475  phplem4dom  6865  fisseneq  6934  nnnninfeq2  7130  archnqq  7419  imasaddfnlemg  12741  ringidss  13218  subrg1  13358  subrgdvds  13362  subrguss  13363  subrginv  13364  islss3  13472  lspsnneg  13512  epttop  13730  metequiv2  14136  limccnpcntop  14284  limccnp2lem  14285  limccnp2cntop  14286  nnsf  14895