Theorem syl5eqss 3316

Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
syl5eqss.1	|- A = B
syl5eqss.2	|- (ph -> B C_ C)

Assertion

Ref	Expression
syl5eqss	|- (ph -> A C_ C)

Proof of Theorem syl5eqss

Step	Hyp	Ref	Expression
1		syl5eqss.2	. 2 |- (ph -> B C_ C)
2		syl5eqss.1	. . 3 |- A = B
3	2	sseq1i 3296	. 2 |- (A C_ C <-> B C_ C)
4	1, 3	sylibr 203	1 |- (ph -> A 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:  syl5eqssr  3317  inss  3485  peano5  4410  spfininduct  4541  fun  5237  fmpt  5693  clos1induct  5881  sbthlem1  6204  spacssnc  6285  frecxp  6315  frecxpg  6316