Theorem sseqtrid 3147

Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

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

Assertion

Ref	Expression
sseqtrid	|- ( ph -> B C_ C )

Proof of Theorem sseqtrid

Step	Hyp	Ref	Expression
1		sseqtrid.2	. 2 |- ( ph -> A = C )
2		sseqtrid.1	. 2 |- B C_ A
3		sseq2 3121	. . 3 |- ( A = C -> ( B C_ A <-> B C_ C ) )
4	3	biimpa 294	. 2 |- ( ( A = C /\ B C_ A ) -> B C_ C )
5	1, 2, 4	sylancl 409	1 |- ( ph -> B C_ C )

Syntax hints:   -> wi 4   = wceq 1331   C_ wss 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084

This theorem is referenced by:  fssdm  5287  fndmdif  5525  fneqeql2  5529  fconst4m  5640  f1opw2  5976  ecss  6470  fopwdom  6730  ssenen  6745  phplem2  6747  fiintim  6817  casefun  6970  caseinj  6974  djufun  6989  djuinj  6991  nn0supp  9029  monoord2  10250  binom1dif  11256  cnpnei  12388  cnntri  12393  cnntr  12394  cncnp  12399  cndis  12410  txdis1cn  12447  hmeontr  12482  hmeoimaf1o  12483  dvcoapbr  12840