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  3758  peano5  4595  xpsspw  4736  iotanul  5190  iotass  5192  fun  5385  fun11iun  5479  fvss  5526  fmpt  5663  fliftrel  5788  opabbrex  5914  1stcof  6159  2ndcof  6160  tfrlemibacc  6322  tfrlemibfn  6324  tfr1onlemssrecs  6335  tfr1onlembacc  6338  tfr1onlembfn  6340  tfrcllemssrecs  6348  tfrcllembacc  6351  tfrcllembfn  6353  caucvgprlemladdrl  7672  peano5nnnn  7886  peano5nni  8916  un0addcl  9203  un0mulcl  9204  strleund  12552  mgmidsssn0  12733  cnptopco  13504  cnconst2  13515  xmetresbl  13722  blsscls2  13775  bj-omtrans  14479