Theorem syl6eqssr 3078

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

Hypotheses

Ref	Expression
syl6eqssr.1	|- ( ph -> B = A )
syl6eqssr.2	|- B C_ C

Assertion

Ref	Expression
syl6eqssr	|- ( ph -> A C_ C )

Proof of Theorem syl6eqssr

Step	Hyp	Ref	Expression
1		syl6eqssr.1	. . 3 |- ( ph -> B = A )
2	1	eqcomd 2094	. 2 |- ( ph -> A = B )
3		syl6eqssr.2	. 2 |- B C_ C
4	2, 3	syl6eqss 3077	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1290   C_ wss 3000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-in 3006  df-ss 3013

This theorem is referenced by:  ffvresb  5475  tposss  6025  sbthlemi5  6724  iooval2  9394  telfsumo  10921  structcnvcnv  11571