Theorem eqsstrrid 3248

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 2211	. 2 |- A = B
3		eqsstrrid.2	. 2 |- ( ph -> B C_ C )
4	2, 3	eqsstrid 3247	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3174

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187

This theorem is referenced by:  abnexg  4511  relcnvtr  5221  resasplitss  5477  fimacnvdisj  5482  fimacnv  5732  f1ompt  5754  tfr1onlemres  6458  tfrcllemres  6471