Theorem syl5sseq 2993

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 2967	. . 3 |- ( A = C -> ( B C_ A <-> B C_ C ) )
4	3	biimpa 280	. 2 |- ( ( A = C /\ B C_ A ) -> B C_ C )
5	1, 2, 4	sylancl 392	1 |- ( ph -> B C_ C )

Syntax hints:   -> wi 4   = wceq 1243   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by:  fndmdif  5272  fneqeql2  5276  fconst4m  5381  f1opw2  5706  ecss  6147  fopwdom  6310  phplem2  6316  nn0supp  8234  monoord2  9236