Theorem 3sstr4d 3229

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 3215	. 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  6450  sucinc2  6513  oawordi  6536  nnnninf  7201  fzoss1  10266  fzoss2  10267  lspss  14033  clsss  14462  ntrss  14463  sslm  14591  txss12  14610  metss2lem  14841  xmettxlem  14853  xmettx  14854  plyss  15082  nnsf  15760  nninfself  15768