Theorem sseqtr4i 3305

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)

Hypotheses

Ref	Expression
sseqtr4.1	⊢ A ⊆ B
sseqtr4.2	⊢ C = B

Assertion

Ref	Expression
sseqtr4i	⊢ A ⊆ C

Proof of Theorem sseqtr4i

Step	Hyp	Ref	Expression
1		sseqtr4.1	. 2 ⊢ A ⊆ B
2		sseqtr4.2	. . 3 ⊢ C = B
3	2	eqcomi 2357	. 2 ⊢ B = C
4	1, 3	sseqtri 3304	1 ⊢ A ⊆ C

Syntax hints:   = wceq 1642   ⊆ 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:  eqimss2i  3327  snsspr1  3857  snsspr2  3858  snsstp1  3859  snsstp2  3860  snsstp3  3861  unissint  3951  iunxdif2  4015  snprss1  4121  opabssxp  4838  dmresi  5005  cnvimass  5017  ssrnres  5060  fvfullfunlem3  5864  mapsspm  6022  sbthlem1  6204