Theorem sseqtrid 3298

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 3272	. . 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 417	1 |- ( ph -> B C_ C )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   C_ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is used by:  fssdm  5552  fndmdif  5817  fneqeql2  5821  fconst4m  5938  f1opw2  6299  fsuppeq  6491  fsuppeqg  6492  ecss  6854  pw2f1odclem  7138  fopwdom  7140  ssenen  7156  phplem2  7158  fiintim  7242  casefun  7431  caseinj  7435  djufun  7450  djuinj  7452  nn0supp  9630  monoord2  10944  binom1dif  12279  ballotfilemro  13324  znleval  15097  cnpnei  15445  cnntri  15450  cnntr  15451  cncnp  15456  cndis  15467  txdis1cn  15504  hmeontr  15539  hmeoimaf1o  15540  dvcoapbr  15933  uhgrspansubgr  16723  vtxdfifiun  16743