Theorem eqsstrid 3174

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 3154	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
4	1, 3	sylibr 133	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1335   ⊆ wss 3102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115

This theorem is referenced by:  eqsstrrid  3175  inss  3337  difsnss  3702  tpssi  3722  peano5  4556  xpsspw  4697  iotanul  5149  iotass  5151  fun  5341  fun11iun  5434  fvss  5481  fmpt  5616  fliftrel  5739  opabbrex  5862  1stcof  6108  2ndcof  6109  tfrlemibacc  6270  tfrlemibfn  6272  tfr1onlemssrecs  6283  tfr1onlembacc  6286  tfr1onlembfn  6288  tfrcllemssrecs  6296  tfrcllembacc  6299  tfrcllembfn  6301  caucvgprlemladdrl  7593  peano5nnnn  7807  peano5nni  8831  un0addcl  9118  un0mulcl  9119  strleund  12265  cnptopco  12609  cnconst2  12620  xmetresbl  12827  blsscls2  12880  bj-omtrans  13518