Theorem syl5sseq 3320

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 3294	. . 3 |- (A = C -> (B C_ A <-> B C_ C))
4	3	biimpa 470	. 2 |- ((A = C /\ B C_ A) -> B C_ C)
5	1, 2, 4	sylancl 643	1 |- (ph -> B C_ C)

Syntax hints:   -> wi 4   = wceq 1642   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  fconst4  5459  ecss  5967  sbthlem3  6206