Theorem sseqtri 3303

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)

Hypotheses

Ref	Expression
sseqtr.1	⊢ A ⊆ B
sseqtr.2	⊢ B = C

Assertion

Ref	Expression
sseqtri	⊢ A ⊆ C

Proof of Theorem sseqtri

Step	Hyp	Ref	Expression
1		sseqtr.1	. 2 ⊢ A ⊆ B
2		sseqtr.2	. . 3 ⊢ B = C
3	2	sseq2i 3296	. 2 ⊢ (A ⊆ B ↔ A ⊆ C)
4	1, 3	mpbi 199	1 ⊢ A ⊆ C

Syntax hints:   = wceq 1642   ⊆ wss 3257

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  sseqtr4i  3304  eqimssi  3325  abssi  3341  ssun2  3427  snprss2  4121  0ima  5014  foimacnv  5303  clos1base  5878  sbthlem1  6203