Theorem sseqtrrd 3223

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 2202	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	1, 3	sseqtrd 3222	1 ⊢ (𝜑 → 𝐴 ⊆ 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:  sseqtrrid  3235  fnfvima  5800  tfrlemiubacc  6397  tfr1onlemubacc  6413  tfrcllemubacc  6426  rdgivallem  6448  nnnninf  7201  nninfwlpoimlemg  7250  dfphi2  12415  ctinf  12674  imasaddfnlemg  13018  imasaddvallemg  13019  subsubm  13187  subsubg  13405  subsubrng  13848  subsubrg  13879  lidlss  14110  toponss  14370  ssntr  14466  iscnp3  14547  cnprcl2k  14550  tgcn  14552  tgcnp  14553  ssidcn  14554  cncnp  14574  txcnp  14615  imasnopn  14643  hmeontr  14657  blssec  14782  blssopn  14829  xmettx  14854  metcnp  14856  plyaddlem1  15091  plymullem1  15092  plycoeid3  15101  nnsf  15760  nninfsellemsuc  15767