Theorem sseqtrrd 3287

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

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

Assertion

Ref	Expression
sseqtrrd	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem sseqtrrd

Step	Hyp	Ref	Expression
1		sseqtrrd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sseqtrrd.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐵)
3	2	eqcomd 2244	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	1, 3	sseqtrd 3286	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1402   ⊆ wss 3220

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  sseqtrrid  3299  fnfvima  5946  tfrlemiubacc  6594  tfr1onlemubacc  6610  tfrcllemubacc  6623  rdgivallem  6645  nnnninf  7459  nninfwlpoimlemg  7508  ccatass  11357  swrdval2  11404  dfphi2  12979  ctinf  13302  imasaddfnlemg  13615  imasaddvallemg  13616  subsubm  13770  subsubg  13980  subsubrng  14498  subsubrg  14529  lidlss  14788  toponss  15053  ssntr  15149  iscnp3  15230  cnprcl2k  15233  tgcn  15235  tgcnp  15236  ssidcn  15237  cncnp  15257  txcnp  15298  imasnopn  15326  hmeontr  15340  blssec  15465  blssopn  15512  xmettx  15537  metcnp  15539  plyaddlem1  15774  plymullem1  15775  plycoeid3  15784  nnsf  16956  nninfsellemsuc  16963