Theorem eqsstrrdi 3200

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 2176	. 2 |- ( ph -> A = B )
3		eqsstrrdi.2	. 2 |- B C_ C
4	2, 3	eqsstrdi 3199	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:  ffvresb  5659  tposss  6225  sbthlemi5  6938  iooval2  9872  telfsumo  11429  structcnvcnv  12432  txss12  13060  txbasval  13061