Theorem syl5eqss 2962

Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
syl5eqss.1	⊢ A = B
syl5eqss.2	⊢ (φ → B ⊆ 𝐶)

Assertion

Ref	Expression
syl5eqss	⊢ (φ → A ⊆ 𝐶)

Proof of Theorem syl5eqss

Step	Hyp	Ref	Expression
1		syl5eqss.2	. 2 ⊢ (φ → B ⊆ 𝐶)
2		syl5eqss.1	. . 3 ⊢ A = B
3	2	sseq1i 2942	. 2 ⊢ (A ⊆ 𝐶 ↔ B ⊆ 𝐶)
4	1, 3	sylibr 137	1 ⊢ (φ → A ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1226   ⊆ wss 2890

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000

This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-in 2897  df-ss 2904

This theorem is referenced by:  syl5eqssr  2963  inss  3139  difsnss  3480  tpssi  3500  peano5  4244  xpsspw  4373  iotanul  4805  iotass  4807  fun  4984  fun11iun  5068  fvss  5110  fmpt  5240  fliftrel  5353  opabbrex  5468  1stcof  5709  2ndcof  5710  tfrlemibacc  5857  tfrlemibfn  5859  peano5set  7301  bj-omtrans  7317