Theorem sseqtrrid 3235

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 3223	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1364   ⊆ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  resdif  5529  fimacnv  5694  tfrlem5  6381  fsumsplit  11591  fprodsplitdc  11780  phimullem  12420  ennnfonelemss  12654  prdssca  12979  prdsbas  12980  prdsplusg  12981  prdsmulr  12982  lspssid  14034  istopon  14335  sscls  14442  mopnfss  14769  plyaddlem1  15069  plymullem1  15070  lgsquadlem2  15405