Theorem sseqtrrdi 3173

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

Hypotheses

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

Assertion

Ref	Expression
sseqtrrdi	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem sseqtrrdi

Step	Hyp	Ref	Expression
1		sseqtrrdi.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sseqtrrdi.2	. . 3 ⊢ 𝐶 = 𝐵
3	2	eqcomi 2158	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	sseqtrdi 3172	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1332   ⊆ wss 3098

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-11 1483  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-in 3104  df-ss 3111

This theorem is referenced by:  iunpw  4434  iotanul  5143  iotass  5145  tfrlem9  6256  tfrlemibfn  6265  tfrlemiubacc  6267  tfrlemi14d  6270  tfr1onlemssrecs  6276  tfr1onlemres  6286  tfrcllemres  6299  exmidfodomrlemr  7116  exmidfodomrlemrALT  7117  uznnssnn  9467  shftfvalg  10695  shftfval  10698  clim2prod  11413  eltopss  12346  difopn  12447  tgrest  12508  txuni2  12595  tgioo  12885