Theorem sseqtrrid 3277

Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

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

Assertion

Ref	Expression
sseqtrrid	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Proof of Theorem sseqtrrid

Step	Hyp	Ref	Expression
1		sseqtrrid.1	. . 3 ⊢ 𝐵 ⊆ 𝐴
2	1	a1i 9	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
3		sseqtrrid.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐴)
4	2, 3	sseqtrrd 3265	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1397   ⊆ wss 3199

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-in 3205  df-ss 3212

This theorem is referenced by:  resdif  5608  fimacnv  5779  tfrlem5  6485  fsumsplit  11991  fprodsplitdc  12180  phimullem  12820  ennnfonelemss  13054  prdssca  13381  prdsbas  13382  prdsplusg  13383  prdsmulr  13384  lspssid  14438  istopon  14766  sscls  14873  mopnfss  15200  plyaddlem1  15500  plymullem1  15501  lgsquadlem2  15836