Theorem sseqtrrdi 3219

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 2193	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	sseqtrdi 3218	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1364   ⊆ wss 3144

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157

This theorem is referenced by:  iunpw  4501  iotanul  5214  iotass  5216  tfrlem9  6348  tfrlemibfn  6357  tfrlemiubacc  6359  tfrlemi14d  6362  tfr1onlemssrecs  6368  tfr1onlemres  6378  tfrcllemres  6391  exmidfodomrlemr  7236  exmidfodomrlemrALT  7237  uznnssnn  9614  shftfvalg  10869  shftfval  10872  clim2prod  11589  reldvdsrsrg  13468  dvdsrvald  13469  dvdsrex  13474  eltopss  13995  difopn  14094  tgrest  14155  txuni2  14242  tgioo  14532