Theorem eqsstrrdi 3195

Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

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

Assertion

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

Proof of Theorem eqsstrrdi

Step	Hyp	Ref	Expression
1		eqsstrrdi.1	. . 3 |- ( ph -> B = A )
2	1	eqcomd 2171	. 2 |- ( ph -> A = B )
3		eqsstrrdi.2	. 2 |- B C_ C
4	2, 3	eqsstrdi 3194	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:  ffvresb  5648  tposss  6214  sbthlemi5  6926  iooval2  9851  telfsumo  11407  structcnvcnv  12410  txss12  12906  txbasval  12907