Theorem eqsstrrid 3275

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 2235	. 2 |- A = B
3		eqsstrrid.2	. 2 |- ( ph -> B C_ C )
4	2, 3	eqsstrid 3274	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3201

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214

This theorem is referenced by:  abnexg  4549  relcnvtr  5263  resasplitss  5524  fimacnvdisj  5529  fimacnv  5784  f1ompt  5806  tfr1onlemres  6558  tfrcllemres  6571