Theorem eqsstrrdi 3236

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 2202	. 2 |- ( ph -> A = B )
3		eqsstrrdi.2	. 2 |- B C_ C
4	2, 3	eqsstrdi 3235	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  ffvresb  5725  tposss  6304  sbthlemi5  7027  iooval2  9990  telfsumo  11631  structcnvcnv  12694  ressbasssd  12747  txss12  14502  txbasval  14503