Theorem syl5sseq 3074

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

Hypotheses

Ref	Expression
syl5sseq.1	|- B C_ A
syl5sseq.2	|- ( ph -> A = C )

Assertion

Ref	Expression
syl5sseq	|- ( ph -> B C_ C )

Proof of Theorem syl5sseq

Step	Hyp	Ref	Expression
1		syl5sseq.2	. 2 |- ( ph -> A = C )
2		syl5sseq.1	. 2 |- B C_ A
3		sseq2 3048	. . 3 |- ( A = C -> ( B C_ A <-> B C_ C ) )
4	3	biimpa 290	. 2 |- ( ( A = C /\ B C_ A ) -> B C_ C )
5	1, 2, 4	sylancl 404	1 |- ( ph -> B C_ C )

Syntax hints:   -> wi 4   = wceq 1289   C_ wss 2999

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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012

This theorem is referenced by:  fssdm  5169  fndmdif  5398  fneqeql2  5402  fconst4m  5509  f1opw2  5842  ecss  6323  fopwdom  6542  ssenen  6557  phplem2  6559  fiintim  6629  casefun  6766  caseinj  6770  djufun  6774  djuinj  6776  nn0supp  8715  monoord2  9893  binom1dif  10868