Theorem eqsstrid 3202

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 3182	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
4	1, 3	sylibr 134	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1353   ⊆ wss 3130

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 3136  df-ss 3143

This theorem is referenced by:  eqsstrrid  3203  inss  3366  difsnss  3739  tpssi  3760  peano5  4598  xpsspw  4739  iotanul  5194  iotass  5196  fun  5389  fun11iun  5483  fvss  5530  fmpt  5667  fliftrel  5793  opabbrex  5919  1stcof  6164  2ndcof  6165  tfrlemibacc  6327  tfrlemibfn  6329  tfr1onlemssrecs  6340  tfr1onlembacc  6343  tfr1onlembfn  6345  tfrcllemssrecs  6353  tfrcllembacc  6356  tfrcllembfn  6358  caucvgprlemladdrl  7677  peano5nnnn  7891  peano5nni  8922  un0addcl  9209  un0mulcl  9210  strleund  12562  mgmidsssn0  12803  cnptopco  13725  cnconst2  13736  xmetresbl  13943  blsscls2  13996  bj-omtrans  14711