Theorem sseqtrid 3234

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 3208	. . 3 |- ( A = C -> ( B C_ A <-> B C_ C ) )
4	3	biimpa 296	. 2 |- ( ( A = C /\ B C_ A ) -> B C_ C )
5	1, 2, 4	sylancl 413	1 |- ( ph -> B 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:  fssdm  5425  fndmdif  5670  fneqeql2  5674  fconst4m  5785  f1opw2  6133  ecss  6644  pw2f1odclem  6904  fopwdom  6906  ssenen  6921  phplem2  6923  fiintim  7001  casefun  7160  caseinj  7164  djufun  7179  djuinj  7181  nn0supp  9318  monoord2  10595  binom1dif  11669  znleval  14285  cnpnei  14539  cnntri  14544  cnntr  14545  cncnp  14550  cndis  14561  txdis1cn  14598  hmeontr  14633  hmeoimaf1o  14634  dvcoapbr  15027