Theorem eqsstrrd 3261

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
eqsstrrd.1	⊢ (𝜑 → 𝐵 = 𝐴)
eqsstrrd.2	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
eqsstrrd	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem eqsstrrd

Step	Hyp	Ref	Expression
1		eqsstrrd.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
2	1	eqcomd 2235	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqsstrrd.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	2, 3	eqsstrd 3260	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1395   ⊆ wss 3197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  ssxpbm  5163  ssxp1  5164  ssxp2  5165  suppssof1  6234  tfrlemiubacc  6474  tfr1onlemubacc  6490  tfrcllemubacc  6503  oaword1  6615  phplem4dom  7019  fisseneq  7092  nnnninfeq2  7292  archnqq  7600  hashdmprop2dom  11061  imasaddfnlemg  13342  resmhm2  13516  ringidss  13987  subrg1  14189  subrgdvds  14193  subrguss  14194  subrginv  14195  islss3  14337  lspsnneg  14378  epttop  14758  metequiv2  15164  limccnpcntop  15343  limccnp2lem  15344  limccnp2cntop  15345  umgredgprv  15909  uspgrupgrushgr  15974  usgrumgruspgr  15977  nnsf  16330