Theorem sseq1 3293

Description: Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Assertion

Ref	Expression
sseq1	⊢ (A = B → (A ⊆ C ↔ B ⊆ C))

Proof of Theorem sseq1

Step	Hyp	Ref	Expression
1		eqss 3288	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
2		sstr2 3280	. . . 4 ⊢ (B ⊆ A → (A ⊆ C → B ⊆ C))
3	2	adantl 452	. . 3 ⊢ ((A ⊆ B ∧ B ⊆ A) → (A ⊆ C → B ⊆ C))
4		sstr2 3280	. . . 4 ⊢ (A ⊆ B → (B ⊆ C → A ⊆ C))
5	4	adantr 451	. . 3 ⊢ ((A ⊆ B ∧ B ⊆ A) → (B ⊆ C → A ⊆ C))
6	3, 5	impbid 183	. 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → (A ⊆ C ↔ B ⊆ C))
7	1, 6	sylbi 187	1 ⊢ (A = B → (A ⊆ C ↔ B ⊆ C))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = 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:  sseq12  3295  sseq1i  3296  sseq1d  3299  nssne2  3329  psseq1  3357  uneqdifeq  3639  sbss  3660  pwjust  3724  elpw  3729  elpwg  3730  pwpw0  3856  sssn  3865  ssunsn2  3866  pwsnALT  3883  unimax  3926  pwadjoin  4120  eqpw1  4163  opkelssetkg  4269  sspw1  4336  sspw12  4337  ssfin  4471  tfinnn  4535  brssetg  4758  iss  5001  fununi  5161  funcnvuni  5162  ffoss  5315  clos1eq1  5875  frd  5923  mapsspm  6022  mapsspw  6023  mapsn  6027  enprmaplem6  6082  ovcelem1  6172  ceex  6175  nclec  6196  lec0cg  6199  ltcpw1pwg  6203  sbthlem2  6205  nc0le1  6217  dflec3  6222  lenc  6224  ce2le  6234  tlenc1c  6241  nchoicelem10  6299  nchoicelem13  6302