Theorem eqsstri 3302

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)

Hypotheses

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

Assertion

Ref	Expression
eqsstri	⊢ A ⊆ C

Proof of Theorem eqsstri

Step	Hyp	Ref	Expression
1		eqsstr.2	. 2 ⊢ B ⊆ C
2		eqsstr.1	. . 3 ⊢ A = B
3	2	sseq1i 3296	. 2 ⊢ (A ⊆ C ↔ B ⊆ C)
4	1, 3	mpbir 200	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:  eqsstr3i  3303  ssrab2  3352  rabssab  3353  difsscompl  3550  pw1ss1c  4159  pw1sspw  4172  opkabssvvki  4210  imagekrelk  4274  dmopabss  4917  resss  4989  rnin  5038  rnxpss  5054  fun0  5155  fnres  5200  f0  5249  fvopab4ndm  5391  ffvresb  5432  isoini2  5499  dmoprabss  5576  dmmptss  5686  ecss  5967  nenpw1pwlem2  6086  sbthlem1  6204  spacssnc  6285  frecxp  6315