Theorem sseqtrrid 3245

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 3233	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1373   ⊆ wss 3167

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-in 3173  df-ss 3180

This theorem is referenced by:  resdif  5551  fimacnv  5716  tfrlem5  6407  fsumsplit  11762  fprodsplitdc  11951  phimullem  12591  ennnfonelemss  12825  prdssca  13151  prdsbas  13152  prdsplusg  13153  prdsmulr  13154  lspssid  14206  istopon  14529  sscls  14636  mopnfss  14963  plyaddlem1  15263  plymullem1  15264  lgsquadlem2  15599