Theorem sseqtrrdi 3253

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 2213	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	sseqtrdi 3252	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1375   ⊆ wss 3177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-11 1532  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-in 3183  df-ss 3190

This theorem is referenced by:  iunpw  4548  iotanul  5270  iotass  5272  tfrlem9  6435  tfrlemibfn  6444  tfrlemiubacc  6446  tfrlemi14d  6449  tfr1onlemssrecs  6455  tfr1onlemres  6465  tfrcllemres  6478  exmidfodomrlemr  7348  exmidfodomrlemrALT  7349  uznnssnn  9740  pfxccatpfx2  11235  shftfvalg  11295  shftfval  11298  clim2prod  12016  reldvdsrsrg  14021  dvdsrvald  14022  dvdsrex  14027  eltopss  14648  difopn  14747  tgrest  14808  txuni2  14895  tgioo  15193  plycoeid3  15396