Theorem eqsstrid 3148

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 3128	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
4	1, 3	sylibr 133	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1332   ⊆ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089

This theorem is referenced by:  eqsstrrid  3149  inss  3311  difsnss  3674  tpssi  3694  peano5  4520  xpsspw  4659  iotanul  5111  iotass  5113  fun  5303  fun11iun  5396  fvss  5443  fmpt  5578  fliftrel  5701  opabbrex  5823  1stcof  6069  2ndcof  6070  tfrlemibacc  6231  tfrlemibfn  6233  tfr1onlemssrecs  6244  tfr1onlembacc  6247  tfr1onlembfn  6249  tfrcllemssrecs  6257  tfrcllembacc  6260  tfrcllembfn  6262  caucvgprlemladdrl  7510  peano5nnnn  7724  peano5nni  8747  un0addcl  9034  un0mulcl  9035  strleund  12086  cnptopco  12430  cnconst2  12441  xmetresbl  12648  blsscls2  12701  bj-omtrans  13325