Theorem eqsstrid 3270

Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Hypotheses

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

Assertion

Ref	Expression
eqsstrid	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem eqsstrid

Step	Hyp	Ref	Expression
1		eqsstrid.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
2		eqsstrid.1	. . 3 ⊢ 𝐴 = 𝐵
3	2	sseq1i 3250	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
4	1, 3	sylibr 134	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:  eqsstrrid  3271  inss  3434  difsnss  3814  tpssi  3837  peano5  4690  xpsspw  4831  iotanul  5294  iotass  5296  fun  5499  fun11iun  5595  fvss  5643  fmpt  5787  fliftrel  5922  ovssunirng  6042  opabbrex  6054  1stcof  6315  2ndcof  6316  tfrlemibacc  6478  tfrlemibfn  6480  tfr1onlemssrecs  6491  tfr1onlembacc  6494  tfr1onlembfn  6496  tfrcllemssrecs  6504  tfrcllembacc  6507  tfrcllembfn  6509  caucvgprlemladdrl  7876  peano5nnnn  8090  peano5nni  9124  un0addcl  9413  un0mulcl  9414  4sqlemafi  12933  4sqlemffi  12934  4sqleminfi  12935  4sqlem11  12939  4sqlem19  12947  strleund  13151  mgmidsssn0  13432  lsptpcl  14373  cnptopco  14911  cnconst2  14922  xmetresbl  15129  blsscls2  15182  perfectlem2  15689  setsvtx  15867  bj-omtrans  16374