Theorem eqsstrrid 3194

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 2174	. 2 |- A = B
3		eqsstrrid.2	. 2 |- ( ph -> B C_ C )
4	2, 3	eqsstrid 3193	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = 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:  abnexg  4431  relcnvtr  5130  resasplitss  5377  fimacnvdisj  5382  fimacnv  5625  f1ompt  5647  tfr1onlemres  6328  tfrcllemres  6341