Theorem syl6sseqr 3046

Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
syl6ssr.1	|- ( ph -> A C_ B )
syl6ssr.2	|- C = B

Assertion

Ref	Expression
syl6sseqr	|- ( ph -> A C_ C )

Proof of Theorem syl6sseqr

Step	Hyp	Ref	Expression
1		syl6ssr.1	. 2 |- ( ph -> A C_ B )
2		syl6ssr.2	. . 3 |- C = B
3	2	eqcomi 2085	. 2 |- B = C
4	1, 3	syl6sseq 3045	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1284   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  iunpw  4229  iotanul  4902  iotass  4904  tfrlem9  5958  tfrlemibfn  5965  tfrlemiubacc  5967  tfrlemi14d  5970  uznnssnn  8665  shftfvalg  9706  shftfval  9709