Theorem 3sstr4d 3228

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 3214	. 2 ⊢ (𝜑 → (𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵))
5	1, 4	mpbird 167	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)

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:  rdgss  6441  sucinc2  6504  oawordi  6527  nnnninf  7192  fzoss1  10247  fzoss2  10248  lspss  13955  clsss  14354  ntrss  14355  sslm  14483  txss12  14502  metss2lem  14733  xmettxlem  14745  xmettx  14746  plyss  14974  nnsf  15649  nninfself  15657