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  3759  peano5  4597  xpsspw  4738  iotanul  5193  iotass  5195  fun  5388  fun11iun  5482  fvss  5529  fmpt  5666  fliftrel  5792  opabbrex  5918  1stcof  6163  2ndcof  6164  tfrlemibacc  6326  tfrlemibfn  6328  tfr1onlemssrecs  6339  tfr1onlembacc  6342  tfr1onlembfn  6344  tfrcllemssrecs  6352  tfrcllembacc  6355  tfrcllembfn  6357  caucvgprlemladdrl  7676  peano5nnnn  7890  peano5nni  8921  un0addcl  9208  un0mulcl  9209  strleund  12561  mgmidsssn0  12802  cnptopco  13692  cnconst2  13703  xmetresbl  13910  blsscls2  13963  bj-omtrans  14678