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  5164  ssxp1  5165  ssxp2  5166  suppssof1  6242  tfrlemiubacc  6482  tfr1onlemubacc  6498  tfrcllemubacc  6511  oaword1  6625  phplem4dom  7031  fisseneq  7107  nnnninfeq2  7307  archnqq  7615  hashdmprop2dom  11079  imasaddfnlemg  13362  resmhm2  13536  ringidss  14007  subrg1  14210  subrgdvds  14214  subrguss  14215  subrginv  14216  islss3  14358  lspsnneg  14399  epttop  14779  metequiv2  15185  limccnpcntop  15364  limccnp2lem  15365  limccnp2cntop  15366  umgredgprv  15930  uspgrupgrushgr  15995  usgrumgruspgr  15998  nnsf  16431