Theorem eqsstrid 3230

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 3210	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
4	1, 3	sylibr 134	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1364   ⊆ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  eqsstrrid  3231  inss  3394  difsnss  3769  tpssi  3790  peano5  4635  xpsspw  4776  iotanul  5235  iotass  5237  fun  5433  fun11iun  5528  fvss  5575  fmpt  5715  fliftrel  5842  ovssunirng  5960  opabbrex  5970  1stcof  6230  2ndcof  6231  tfrlemibacc  6393  tfrlemibfn  6395  tfr1onlemssrecs  6406  tfr1onlembacc  6409  tfr1onlembfn  6411  tfrcllemssrecs  6419  tfrcllembacc  6422  tfrcllembfn  6424  caucvgprlemladdrl  7764  peano5nnnn  7978  peano5nni  9012  un0addcl  9301  un0mulcl  9302  4sqlemafi  12591  4sqlemffi  12592  4sqleminfi  12593  4sqlem11  12597  4sqlem19  12605  strleund  12808  mgmidsssn0  13088  lsptpcl  14028  cnptopco  14566  cnconst2  14577  xmetresbl  14784  blsscls2  14837  perfectlem2  15344  bj-omtrans  15710