Theorem eqsstrid 3143

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 3123	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
4	1, 3	sylibr 133	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1331   ⊆ wss 3071

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084

This theorem is referenced by:  eqsstrrid  3144  inss  3306  difsnss  3666  tpssi  3686  peano5  4512  xpsspw  4651  iotanul  5103  iotass  5105  fun  5295  fun11iun  5388  fvss  5435  fmpt  5570  fliftrel  5693  opabbrex  5815  1stcof  6061  2ndcof  6062  tfrlemibacc  6223  tfrlemibfn  6225  tfr1onlemssrecs  6236  tfr1onlembacc  6239  tfr1onlembfn  6241  tfrcllemssrecs  6249  tfrcllembacc  6252  tfrcllembfn  6254  caucvgprlemladdrl  7486  peano5nnnn  7700  peano5nni  8723  un0addcl  9010  un0mulcl  9011  strleund  12047  cnptopco  12391  cnconst2  12402  xmetresbl  12609  blsscls2  12662  bj-omtrans  13154