Theorem syl5eqss 2983

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 2963	. 2 ⊢ (A ⊆ 𝐶 ↔ B ⊆ 𝐶)
4	1, 3	sylibr 137	1 ⊢ (φ → A ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1242   ⊆ wss 2911

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925

This theorem is referenced by:  syl5eqssr  2984  inss  3160  difsnss  3501  tpssi  3521  peano5  4264  xpsspw  4393  iotanul  4825  iotass  4827  fun  5006  fun11iun  5090  fvss  5132  fmpt  5262  fliftrel  5375  opabbrex  5491  1stcof  5732  2ndcof  5733  tfrlemibacc  5881  tfrlemibfn  5883  peano5nni  7678  un0addcl  7971  un0mulcl  7972  peano5set  9374  bj-omtrans  9390