Theorem eqsstrrdi 3246

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 2211	. 2 |- ( ph -> A = B )
3		eqsstrrdi.2	. 2 |- B C_ C
4	2, 3	eqsstrdi 3245	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179

This theorem is referenced by:  ffvresb  5743  tposss  6332  sbthlemi5  7063  iooval2  10037  telfsumo  11777  structcnvcnv  12848  ressbasssd  12901  txss12  14738  txbasval  14739