Theorem 3sstr4d 3273

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
3sstr4d.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3sstr4d.2	⊢ (𝜑 → 𝐶 = 𝐴)
3sstr4d.3	⊢ (𝜑 → 𝐷 = 𝐵)

Assertion

Ref	Expression
3sstr4d	⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Proof of Theorem 3sstr4d

Step	Hyp	Ref	Expression
1		3sstr4d.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		3sstr4d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐴)
3		3sstr4d.3	. . 3 ⊢ (𝜑 → 𝐷 = 𝐵)
4	2, 3	sseq12d 3259	. 2 ⊢ (𝜑 → (𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵))
5	1, 4	mpbird 167	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Syntax hints:   → wi 4   = wceq 1398   ⊆ wss 3201

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214

This theorem is referenced by:  ressuppss  6432  suppfnss  6435  suppssfvg  6441  rdgss  6592  sucinc2  6657  oawordi  6680  nnnninf  7368  fzoss1  10453  fzoss2  10454  swrd0g  11290  lspss  14478  clsss  14912  ntrss  14913  sslm  15041  txss12  15060  metss2lem  15291  xmettxlem  15303  xmettx  15304  plyss  15532  ifpsnprss  16267  nnsf  16714  nninfself  16722