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  6855  pw2f1odclem  7140  fopwdom  7142  ssenen  7158  phplem2  7160  fiintim  7244  casefun  7431  caseinj  7435  djufun  7450  djuinj  7452  nn0supp  9628  monoord2  10943  binom1dif  12278  ballotfilemro  13323  znleval  15096  mplsubgfileminv  15233  cnpnei  15462  cnntri  15467  cnntr  15468  cncnp  15473  cndis  15484  txdis1cn  15521  hmeontr  15556  hmeoimaf1o  15557  dvcoapbr  15950  uhgrspansubgr  16740  vtxdfifiun  16760