Theorem sseqtrrdi 3196

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 2174	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	sseqtrdi 3195	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1348   ⊆ wss 3121

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134

This theorem is referenced by:  iunpw  4465  iotanul  5175  iotass  5177  tfrlem9  6298  tfrlemibfn  6307  tfrlemiubacc  6309  tfrlemi14d  6312  tfr1onlemssrecs  6318  tfr1onlemres  6328  tfrcllemres  6341  exmidfodomrlemr  7179  exmidfodomrlemrALT  7180  uznnssnn  9536  shftfvalg  10782  shftfval  10785  clim2prod  11502  eltopss  12801  difopn  12902  tgrest  12963  txuni2  13050  tgioo  13340