Theorem syl6eqss 3077

Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
syl6eqss.1	⊢ (𝜑 → 𝐴 = 𝐵)
syl6eqss.2	⊢ 𝐵 ⊆ 𝐶

Assertion

Ref	Expression
syl6eqss	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem syl6eqss

Step	Hyp	Ref	Expression
1		syl6eqss.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		syl6eqss.2	. . 3 ⊢ 𝐵 ⊆ 𝐶
3	2	a1i 9	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	1, 3	eqsstrd 3061	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1290   ⊆ wss 3000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-in 3006  df-ss 3013

This theorem is referenced by:  syl6eqssr  3078  resasplitss  5203  fimacnv  5442  en2other2  6883  exmidfodomrlemim  6888  toponsspwpwg  11781  ntrss2  11882  bj-nntrans  12119  nninfsellemsuc  12176