Theorem eqsstrid 3193

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 3173	. 2 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)
4	1, 3	sylibr 133	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1348   ⊆ wss 3121

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134

This theorem is referenced by:  eqsstrrid  3194  inss  3357  difsnss  3726  tpssi  3746  peano5  4582  xpsspw  4723  iotanul  5175  iotass  5177  fun  5370  fun11iun  5463  fvss  5510  fmpt  5646  fliftrel  5771  opabbrex  5897  1stcof  6142  2ndcof  6143  tfrlemibacc  6305  tfrlemibfn  6307  tfr1onlemssrecs  6318  tfr1onlembacc  6321  tfr1onlembfn  6323  tfrcllemssrecs  6331  tfrcllembacc  6334  tfrcllembfn  6336  caucvgprlemladdrl  7640  peano5nnnn  7854  peano5nni  8881  un0addcl  9168  un0mulcl  9169  strleund  12506  mgmidsssn0  12638  cnptopco  13016  cnconst2  13027  xmetresbl  13234  blsscls2  13287  bj-omtrans  13991