Theorem syl5eqss 3315

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

Hypotheses

Ref	Expression
syl5eqss.1	⊢ A = B
syl5eqss.2	⊢ (φ → B ⊆ C)

Assertion

Ref	Expression
syl5eqss	⊢ (φ → A ⊆ C)

Proof of Theorem syl5eqss

Step	Hyp	Ref	Expression
1		syl5eqss.2	. 2 ⊢ (φ → B ⊆ C)
2		syl5eqss.1	. . 3 ⊢ A = B
3	2	sseq1i 3295	. 2 ⊢ (A ⊆ C ↔ B ⊆ C)
4	1, 3	sylibr 203	1 ⊢ (φ → A ⊆ C)

Syntax hints:   → wi 4   = wceq 1642   ⊆ wss 3257

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  syl5eqssr  3316  inss  3484  peano5  4409  spfininduct  4540  fun  5236  fmpt  5692  clos1induct  5880  sbthlem1  6203  spacssnc  6284  frecxp  6314  frecxpg  6315