Theorem eqsstrrdi 3301

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 2244	. 2 |- ( ph -> A = B )
3		eqsstrrdi.2	. 2 |- B C_ C
4	2, 3	eqsstrdi 3300	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1402   C_ wss 3220

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  ffvresb  5865  tposss  6510  sbthlemi5  7271  iooval2  10299  telfsumo  12214  structcnvcnv  13349  ressbasssd  13403  txss12  15293  txbasval  15294