Theorem sseq1i 3296

Description: An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
sseq1i.1	⊢ A = B

Assertion

Ref	Expression
sseq1i	⊢ (A ⊆ C ↔ B ⊆ C)

Proof of Theorem sseq1i

Step	Hyp	Ref	Expression
1		sseq1i.1	. 2 ⊢ A = B
2		sseq1 3293	. 2 ⊢ (A = B → (A ⊆ C ↔ B ⊆ C))
3	1, 2	ax-mp 5	1 ⊢ (A ⊆ C ↔ B ⊆ C)

Syntax hints:   ↔ wb 176   = 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:  eqsstri  3302  syl5eqss  3316  ssab  3337  rabss  3344  uniiunlem  3354  prss  3862  prssg  3863  sstp  3871  tpss  3872  iunss  4008  cores2  5092