Theorem eqsstrid 3201

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 3181	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
4	1, 3	sylibr 134	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1353   ⊆ wss 3129

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142

This theorem is referenced by:  eqsstrrid  3202  inss  3365  difsnss  3738  tpssi  3759  peano5  4596  xpsspw  4737  iotanul  5192  iotass  5194  fun  5387  fun11iun  5481  fvss  5528  fmpt  5665  fliftrel  5790  opabbrex  5916  1stcof  6161  2ndcof  6162  tfrlemibacc  6324  tfrlemibfn  6326  tfr1onlemssrecs  6337  tfr1onlembacc  6340  tfr1onlembfn  6342  tfrcllemssrecs  6350  tfrcllembacc  6353  tfrcllembfn  6355  caucvgprlemladdrl  7674  peano5nnnn  7888  peano5nni  8918  un0addcl  9205  un0mulcl  9206  strleund  12554  mgmidsssn0  12735  cnptopco  13593  cnconst2  13604  xmetresbl  13811  blsscls2  13864  bj-omtrans  14568