Theorem eqsstrid 3188

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 3168	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
4	1, 3	sylibr 133	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1343   ⊆ wss 3116

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  eqsstrrid  3189  inss  3352  difsnss  3719  tpssi  3739  peano5  4575  xpsspw  4716  iotanul  5168  iotass  5170  fun  5360  fun11iun  5453  fvss  5500  fmpt  5635  fliftrel  5760  opabbrex  5886  1stcof  6131  2ndcof  6132  tfrlemibacc  6294  tfrlemibfn  6296  tfr1onlemssrecs  6307  tfr1onlembacc  6310  tfr1onlembfn  6312  tfrcllemssrecs  6320  tfrcllembacc  6323  tfrcllembfn  6325  caucvgprlemladdrl  7619  peano5nnnn  7833  peano5nni  8860  un0addcl  9147  un0mulcl  9148  strleund  12483  mgmidsssn0  12615  cnptopco  12862  cnconst2  12873  xmetresbl  13080  blsscls2  13133  bj-omtrans  13838