Theorem eqsstrrid 3189

Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Hypotheses

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

Assertion

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

Proof of Theorem eqsstrrid

Step	Hyp	Ref	Expression
1		eqsstrrid.1	. . 3 |- B = A
2	1	eqcomi 2169	. 2 |- A = B
3		eqsstrrid.2	. 2 |- ( ph -> B C_ C )
4	2, 3	eqsstrid 3188	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1343   C_ wss 3116

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  abnexg  4424  relcnvtr  5123  resasplitss  5367  fimacnvdisj  5372  fimacnv  5614  f1ompt  5636  tfr1onlemres  6317  tfrcllemres  6330